Important Trigonometric Functions Questions for CBSE Class 11th

In a triangle A B C , the lengths of sides A C and A B are 12cm and 5 cm respectively. If the area of triagnle A B C is 30 cm 2 and R and r are respectively the radii of circumcircle and incircle of triangle A B C then the value of 2 R + r is equal to

The area of the right angled triangle interms of its circumradius and inradius is

Given A = sin 2 ⁡ θ + cos 4 ⁡ θ , then for all real θ

∑ r = 1 n − 1 cos 2 ⁡ r π n is equal to

The value of 16 sin 44° sin 108° sin 72° sin36° is equal to

If the equation x 2 + 5 + 4 cos α + β = 2 x has at least one solution where α , β ∈ 2 , 5 then the value of α + β is equal to

If 3 sin π x + cos π x = x 2 − 2 3 x + 19 9 , then x is equal to

Let n be a positive integer such that sin π 2 n + cos π 2 n = n 2 then n =

The number of solutions of sec x − tan x = 3 , x ∈ [ 0 , 3 π ] is

If cos θ = 1 − 2 t 2 , 0 ∘ < t < 180 ∘ and cos 50 ∘ = t then the value of θ is

If a flag-staff of 6 meters height placed on the top of a tower throws a shadow of 2 3 meters along the ground then the angle (in degrees) that the sun makes with the ground is

If α , β , γ , and δ are in arithmetic progression. Then sin α + β + γ + δ ≠

A bird is sitting on the top of a vertical pole 20m high and its elevation from a point O on the ground is 45 0 . It flies off horizontally straight away from the point O. After one second, the elevation of bird from O is reduced to 30 0 , then the speed (in m/s), of the bird is

AB is a vertical pole and C is its middle point. The end A is on level ground and P is any point on the level ground other than A. The portion CB subtends an angle β at P. If AP: AB = 2 : 1, and tan β = 2 k , t h e n k =

The angle of elevation of top of an unfinished tower at a point distance 120m from its base is 45 0 . How much height must the tower be raise so that angle of elevation of top at the same point to be 60 0 .

If C o s 20 0 = k and cos x = 2 k 2 − 1 , then the possible values of x between 0 0 and 360 0 are

If x is real, cos θ = x + 1 x then

If sec ⁡ x + sec 2 ⁡ x = 1 then the value of tan 8 ⁡ x − tan 4 ⁡ x − 2 tan 2 ⁡ x + 1 is equal to

3 cosec ⁡ 20 ∘ − sec ⁡ 20 ∘ is equal to

sin ⁡ 47 ∘ + sin ⁡ 61 ∘ − sin ⁡ 11 ∘ − sin ⁡ 25 ∘ is equal to

If tan ⁡ A + sin ⁡ A = m and tan ⁡ A − sin ⁡ A = n then m 2 − n 2 2 mn is equal to

If tan ⁡ θ , 2 tan ⁡ θ + 2 and 3 tan ⁡ θ + 3 are in GP, then the value of 7 − 5 cot ⁡ θ 9 − 4 sec 2 ⁡ θ − 1 is

lf A,B,C and D are the angles of a cyclic quadrilateral, then cos A + cos B + cos C + cos D is equal to

If cos ⁡ θ = 1 2 x + 1 x then 1 2 x 2 + 1 x 2 is equal to

If 3 − tan 2 ⁡ π 7 1 − tan 2 ⁡ π 7 = λcos ⁡ π 7 , then λ is .

The set of all possible values of α in [ − π , π ] such that 1 − sin ⁡ α 1 + sin ⁡ α is equal to sec ⁡ α − tan ⁡ α , is

lf A, B, C are angles of a triangle, then 2 sin ⁡ A 2 cosec ⁡ B 2 sin ⁡ C 2 − sin ⁡ Acot ⁡ B 2 − cos ⁡ A is

If y = ( 1 + tan ⁡ A ) ( 1 − tan ⁡ B ) where A − B = π 4 ,then ( y + 1 ) y + 1 is equal to

If 15 sin 4 ⁡ α + 10 cos 4 ⁡ α = 6 then the value of 8 cosec 6 ⁡ α − 27 sec 6 ⁡ α is

Number of solutions of the equation 4 sin 2 ⁡ x + tan 2 ⁡ x + cot 2 ⁡ x + cosec 2 ⁡ x = 6 in [ 0 , π ] is

If cos ⁡ A + cos ⁡ B = m and sin ⁡ A + sin ⁡ B = n where m , n ≠ 0 , then sin ⁡ ( A + B ) is equal to

In a cyclic quadrilateral ABCD , the value of cos ⁡ A + cos ⁡ B + cos ⁡ C + cos ⁡ D is

If p 1 , p 2 , p 3 are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then cos ⁡ A p 1 + cos ⁡ B p 2 + cos ⁡ C p 3 is equal to

If A + B + C = π , n ∈ Z , then tan ⁡ n A + tan ⁡ n B + tan ⁡ n C is equal

If the radius of the in circle of a triangle with its sides 5 k , 6 k , and 5 k is 6, then k is equal to

If sin ⁡ α + cos ⁡ α = m , then sin 6 ⁡ α + cos 6 ⁡ α is equal to

If 1 + sin ⁡ x + sin 2 ⁡ x + sin 3 ⁡ x + … + … ∞ is equal to 4 + 2 3 , 0 < x < π , then x =

If sin ⁡ ( x − y ) = cos ⁡ ( x + y ) = 1 2 the values of x and y flying between 0° and 90° are given by

In any ∆ A B C , the value of a b 2 + c 2 cos ⁡ A + b c 2 + a 2 cos ⁡ B + c a 2 + b 2 cos ⁡ C =

If 0 ≤ x ≤ π and 81 sin 2 ⁡ x + 81 cos 2 ⁡ x = 30 , then x is equal to

If △ A B C , 8 R 2 = a 2 + b 2 + c 2 , then the triangle A B C , is

If sin ⁡ A + sin ⁡ B = a and cos ⁡ A + cos ⁡ B = b , then cos ⁡ ( A + B )

If y = sin ⁡ 3 θ sin ⁡ θ , θ ≠ n π then

If in a triangle A B C , right angled at B , s – a = 3 , s – c = 2 , then a 2 + c 2 =

If sec α and cosec α are the roots of the equation x 2 − a x + b = 0 , then

tan ⁡ 2 π 5 − tan ⁡ π 15 − 3 tan ⁡ 2 π 5 tan ⁡ π 15 is equal to

The value of cos ⁡ π 15 cos ⁡ 2 π 15 cos ⁡ 3 π 15 cos ⁡ 4 π 15 cos ⁡ 5 π 15 cos ⁡ 6 π 15 cos ⁡ 7 π 15 , is

The maximum value of cos 2 ⁡ π 3 − x − cos 2 ⁡ π 3 + x is

The value of cos ⁡ 10 ∘ − sin ⁡ 10 ∘ , is

In a △ A B C , a = 2 b and A = 3 B , then A =

The expression tan 2 ⁡ α + cot 2 ⁡ α , is

The value of cot ⁡ θ − tan ⁡ θ − 2 tan ⁡ 2 θ − 4 tan ⁡ 4 θ − 8 cot ⁡ 8 θ is

If y = sec 2 ⁡ θ − tan ⁡ θ sec 2 ⁡ θ + tan ⁡ θ , then

The value of tan ⁡ 6 ∘ tan ⁡ 42 ∘ tan ⁡ 66 ∘ tan ⁡ 78 ∘ , is

The maximum value of 5 cos ⁡ θ + 3 cos ⁡ θ + π 3 + 3 is

In 0 , π 2 , one solution of 3 + 1 cos e c x + 3 − 1 sec x = 4 2 is 5 π 12 . If other solution is λ π 36 then λ =

If 3 sec 4 x + 6 tan 4 x = 2 , then solution set is

If x , y ∈ 0 , 15 , then the number of solutions x , y of the equation 3 cos e c 2 x − 1 × 4 y 2 − 4 y + 2 ≤ 1 is

If sin θ cos 3 θ + sin 3 θ cos 9 θ + sin 9 θ cos 27 θ + sin 27 θ cos 81 θ = A tan B θ − tan C θ then B − C A = A > 0 , θ ≠ 2 n + 1 π 2 , n ∈ Z

Let θ ∈ [ 0 , 4 π ] satisies the equation ( cos θ + 1 ) ( cos θ + 2 ) ( cos θ + 3 ) = 24 , then the sum of all the values of θ is of the form k π then K =

If 2 tan 2 x − 5 sec x = 1 for exactly 10 distinct values of x ∈ 0 , n π , n ∈ N then the greatest value of n is

The general solution of the equation 1 − sin x + …. + − 1 n sin n x + …. 1 + sin x + …. + sin n x + ….. = 1 − cos 2 x 1 + cos 2 x , x ≠ 2 n + 1 π / 2 , n ∈ I is

The general solution of the equation 2 cot θ 2 = 1 + cot θ 2 is

The equation cos p − 1 x 2 + cos p x + sin p = 0 in the variable x has real roots. Then p can take any value in the interval

The number of solutions of 2 sin 2 x 2 sin 2 x = x 2 + 1 x 2 for 0 < x ≤ π 2

If sin 3 α = 4 sin α sin ( x + α ) sin ( x − α ) then

If sin θ + cos e c θ = 2 , then sin n θ + cos e c n θ is equal to

sec 2 θ − tan θ sec 2 θ + tan θ l i e s b e t w e e n b , a , t h e n 3 b + a =

A flagstaff stands vertically on a pillar, the height of the flagstaff being double the height of the pillar. A man on the ground at a distance finds that both the pillar and the flagstaff subtend equal angles at his eyes. The ratio of the height of the pillar and the distance of the man from the pillar, is

sin θ = 1 2 x y + y x necessarily implies

T h e n u m b e r o f v a l u e s o f x i n t h e i n t e r v a l 0 , 3 π s a t i s f y i n g t h e e q u a t i o n 2 sin 2 x + 5 sin x − 3 = 0 i s

The angle of elevation of an object from a point P on the level ground is α . Moving d meters on the ground towards the object, the angle of elevation is found to be β . Then the height (in meters) of the object is

From the top of a building h meters high, the angle of depression of an object on the ground is α . The distance (in meter) of the object from the foot of the building is

The angles of elevation measured from two points A and B on a horizontal line from the foot of a tower are α and β . If AB=d, then the height of the tower is

As seen from the top of fort of height a” meters the angle of depression of the upper and the lower end of a lamp post are α and β respectively. The height of the lamp post is

From the top of a light house, the angles of depression of two stations on opposite side of it at a distance ‘a’ apart are α and β . The height of the light house is

The top of a hill observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. The height of the hill is

A tower, of x meters high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant ‘y’ meters from the foot of the tower. Then the length of the flagstaff in meters is

B is a point situated vertically upwards on A at a distance of a unit. C is a point at a height ‘2a’ units from a horizontal passing through A and the angles of elevation of C from A and B are α and β respectively, then cot α tan β =

A man observes that the tops of three poles standing in front of him are in a line. If the heights of the poles are in A.P. then the horizontal distances of the poles from the man are in

The height of the tower vertically at the orthocenter of the triangle (of sides a, b, c) and subtending ∠ A at A is

3 + cot 76 ° cot 16 ° cot 76 ° + cot 16 ° is equal to

For the equation 1 − 2 x − x 2 = tan 2 x + y + cot 2 x + y

The solution of the equation cos 103 x − sin 103 x = 1 are

If x + y = 2 π 3 a n d sinx sin y = 2 then

The measure of an angle in degrees, grades and radians be D,G and C respectively then the relation between them D 90 = G 100 = 2 C π b u t 1 c = 180 π ° = 57 ° .17 ‘ 44.8 ” then the value of cos 1 c is

A quadratic equation whose roots are cos e c 2 θ and sec 2 θ can be

a sin x = b cos x = 2 c tan x 1 − tan 2 x and a 2 − b 2 2 = k c 2 a 2 + b 2 ⇒ k =

I n a Δ P Q R , i f 3 sin P + 4 cos Q = 6 a n d 4 sin Q + 3 cos P = 1 then the acute angle R is equal to

If for real values of x , cos θ = x + 1 x then

If x = h + p sec α , y = k + q cos e c α then p x − h 2 + q y − k 2 =

a 2 cos 2 2 π 3 − 4 a 2 tan 2 3 π 4 + 2 a 2 sin 2 2 π 3 =

If 0 ≤ x ≤ π 2 , 4 sin 2 x + 4 cos 2 x = 5 then x =

If the equation k cos x − 3 sin x = k + 1 has a solution for x then

I f 0 < θ < π 2 , tan θ = cos 29 ° + sin 29 ° cos 29 ° − sin 29 ° , t h e n θ =

Number of solutions of cos x 5 + sin x 3 = 1 in 0 , 2 π

The maximum distance of a point on the graph of the function y = 3 sin x + cos x from x -axis is

I f 0 ≤ a ≤ 3 , 0 ≤ b ≤ 3 a n d t h e e q u a t i o n x 2 + 4 + 3 cos ( a x + b ) = 2 x h a s a t l e a s t s o l u t i o n . T h e n t h e v a l u e s o f a + b i s

In a triangle ABC, angle A is greater than B, if the measures of angle A and B satisfy the equation 3 sin x − 4 sin 3 x − k = 0 , 0 < K < 1 Then the measure of angle C is

sin 1 0 + sin 2 0 + sin 3 0 + sin 4 0 cos 1 0 + cos 2 0 + cos 3 0 + cos 4 0 =

cos 28 0 + sin 28 0 = k 3 then cos 17 0 =

In Δ A B C cos A cos B cos C = 3 – 1 8 and sin A sin B sin C = 3 + 3 8 , then the value of tan A + tan B + tan C =

I f x i > 0 , 1 ≤ i ≤ n , and x 1 + x 2 + x 3 … + x n = π , then the greatest value of sin x 1 + sin x 2 + sin x 3 + … + sin x n =

If α , β , γ ∈ 0, π 2 then the value of sin α + β + γ sin α + sin β + sin γ is

sin x + cos x = (where {.} and [.] are the fractional par and the integral part

If the quadratic equation whole roots are cos e c 2 θ , s e c 2 θ is x 2 – λ x + λ = 0 then

The value of the function sgn ( cos x – sin x ) in x ∈ π 4 , π 2 = ( sgn is Signum function)

A = π 7 B = 2 π 7 C = 4 π 7 and cos A cos B cos C = – 1 8 then ∑ tan A . tan B =

θ = 2 π 2009 then cos θ cos 2 θ cos 3 θ … … . . cos 1004 θ

If sin x + cos x + tan x + cosec x + sec x + cot x = 7 , then sin 2 x =

a sin 2 x + b cos 2 x = c , b sin 2 y + a cos 2 y = d and a tan x = b tan y then a 2 b 2 =

f ( x ) = sin x + sin 3 x + sin 5 x + … + sin 15 x cos x + cos 3 x + cos 5 x + … + cos 15 x then the value of f ( x ) at x = π 32

If y = ( sin x + cosec x ) 2 + ( cos x + sec x ) 2 then min value of y =

Min. value of 1 3 sin θ – 4 cos θ + 7

The maximum value of 3 cos x + 4 sin x + 5 is

If A + B + C = π then the minimum value of tan 2 A 2 + tan 2 B 2 + tan 2 C 2

The values of x in which sin x – cos x is defined in [ 0 , 2 π ]

Which of the following is possible?

The min. value of 2 sin x + 2 cos x =

The least value of sec A + sec B + sec C in a acute angled triangle

For any real θ the max value of cos 2 ( cos θ ) + sin 2 ( sin θ ) =

The equation cos 8 x + b cos 4 x + 1 = 0 will have solution if b ∈

If a sin x + b cos ( x + θ ) + b cos ( x – θ ) = d , then min value of | cos θ | =

The number of solutions of the equation 1 + sin 4 x = cos 2 3 x , x ∈ – 5 π 2 , 5 π 2 is

The number of solutions of the equation sin 2 x 2 · sin x + 1 = 0 is

Number of solutions of the equation cos 2 x + 3 + 1 2 sin x = 3 4 + 1 in the interval [ – π , π ] is

A vertical pole stands at a point A on the boundary of a circular park of radius ‘a’ and subtends an angle β at another point B on the boundary. If the chord AB subtends an angle α at the center of the park, the height of the pole is

A flag staff A B 4m high is placed on a vertical tower. The angle of elevation of foot of the flag staff B at two points D , E on the ground level and lying on either side of foot of the tower are α and 60 0 respectively with 4m and 2 3 m as the respective distances of these points from the foot of tower. The angle subtended by the flag staff at point D is

If in a Δ A B C , b ( b + c ) = a 2 and c ( c + a ) = b 2 , then cos ⁡ A ⋅ cos ⁡ B ⋅ cos ⁡ C =

Number of solution(s) of the equation sin 2 θ + cos ⁡ 2 θ = − 1 2 , θ ∈ 0 , π 2 , is

The sides of a triangle ABC satisfy the relations a+b-c=2 and 2ab-c 2 = 4 . t hen square of the area of triangle is

If cos ⁡ A + cos ⁡ B + cos ⁡ C = 0 and cos ⁡ 3 A + cos ⁡ 3 B + cos ⁡ 3 C = λ cos ⁡ A cos ⁡ B cos ⁡ C then λ is

Solution of the equation 3 3 sin 3 ⁡ x + cos 3 ⁡ x + 3 3 sin ⁡ x cos ⁡ x = 1

The minimum value of 27 cos ⁡ x + 81 sin ⁡ x is equal to

If S i n π C o t θ 4 = C o s π T a n θ 4 and θ is in the first quadrant then θ =—–

If sin 4 ⁡ x 2 + cos 4 ⁡ x 3 = 1 5 then

The solution of tan ⁡ x + tan ⁡ 2 x + tan ⁡ 3 x = tan ⁡ xtan ⁡ 2 xtan ⁡ 3 x is

The solution of the equation kcos ⁡ x − 3 sin ⁡ x = k + 1 is possible if

If cos ⁡ α + cos ⁡ β = 0 = sin ⁡ α + sin ⁡ β then cos ⁡ 2 α + cos ⁡ 2 β is equal to

Which one of the following number (s) is/are rational?

The numerical value of cos π 9 cos 2 π 9 cos 3 π 9 cos 4 π 9 is equal to

The minimum vertical distance between the graphs of y=2+ sinx and y= cosx is

If cos ⁡ θ 1 = 2 cos ⁡ θ 2 , then tan ⁡ θ 1 − θ 2 2 tan ⁡ θ 1 + θ 2 2

The value of sin ⁡ 1 ∘ + sin ⁡ 3 ∘ + sin ⁡ 5 ∘ + sin ⁡ 7 ∘ cos ⁡ 1 ∘ ⋅ cos ⁡ 2 ∘ ⋅ sin ⁡ 4 ∘ is .

If sin ⁡ θ 1 sin ⁡ θ 2 − cos ⁡ θ 1 cos ⁡ θ 2 + 1 = 0 then the value of tan ⁡ θ 1 / 2 cot ⁡ θ 2 / 2 is equal to

Value of 3 + cot ⁡ 80 ∘ cot ⁡ 20 ∘ cot ⁡ 80 ∘ + cot ⁡ 20 ∘ is equal to

If cos ⁡ A = 3 / 4 , then the value of 16 cos 2 ⁡ ( A / 2 ) − 32 sin ⁡ ( A / 2 ) sin ⁡ ( 5 A / 2 ) is

If x = a cos 3 θ sin 2 θ , y = a sin 3 θ cos 2 θ and x 2 + y 2 p ( x y ) q ( p , q , ∈ N ) is independent of θ , then

sin 2 ⁡ π 8 + A 2 − sin 2 ⁡ π 8 − A 2 is equal to

If A + B + C = π / 2 , then sin 2 ⁡ A + sin 2 ⁡ B + sin 2 ⁡ C + 2 sin ⁡ A sin ⁡ B sin ⁡ C is equal to

If 0 < θ < π / 8 , then 2 + 2 + 2 cos ⁡ ( 4 θ ) is equal to

The greatest value of f ( x ) = 2 sin x + sin 2 x on [ 0 , 3 π / 2 ] , is given b

A solution ( x , y ) of x 2 + 2 x sin ⁡ x y + 1 = 0 is

Let f ( θ ) = cos ⁡ θ cos ⁡ 2 θ cos ⁡ 4 θ cos ⁡ 7 θ , then f ( π / 15 ) is equal to

Number of values of x lying in the interval [ 0 , 5 π ] and satisfying the equation sin ⁡ x = tan ⁡ x is.

If 2 cos ⁡ x + 2 cos ⁡ 3 x = cos ⁡ y , 2 sin ⁡ x + 2 sin ⁡ 3 x = sin ⁡ y then the value of cos ⁡ 2 x of

If cos ⁡ ( α + β ) = 4 / 5 and sin ⁡ ( α − β ) = 5 / 13 where 0 ≤ α , β ≤ π / 4 , then tan ⁡ 2 α =

If 3 sin ⁡ 2 θ 5 + 4 cos ⁡ 2 θ = 1 then the value of tan θ is equal to

If A,B,C are the angles of a triangle and c os ⁡ B + cos ⁡ C = 4 sin 2 A 2 , then tan ⁡ B 2 tan ⁡ C 2 is equal to

The number of values of x in the interval [ 0 , 3 π ] satisfying the equation 2 sin 2 ⁡ x + 5 sin ⁡ x − 3 = 0 is

If x = a sec 3 ⁡ θ tan ⁡ θ , y = b tan 3 ⁡ θ sec ⁡ θ , then sin 2 ⁡ θ is equal to

If cos ⁡ α + cos ⁡ β = 0 = sin ⁡ α + sin ⁡ β , then cos ⁡ 2 α + cos ⁡ 2 β =

If cos ⁡ θ = cos ⁡ α cos ⁡ β , then tan ⁡ θ + α 2 tan ⁡ θ − α 2 is equal to

If 0 < A < π 6 and sin ⁡ A + cos ⁡ A = 7 2 , then tan ⁡ A 2 =

The value of sin ⁡ 10 ∘ + sin ⁡ 20 ∘ + sin ⁡ 30 ∘ + … + sin ⁡ 360 ∘ , is

If cos ⁡ A = 3 4 , then the value of sin ⁡ A 2 sin ⁡ 5 A 2 , is

The angle of a right angled triangle are A . P . The ratio of the in-radius and the perimeter is

If tan ⁡ α 2 and tan ⁡ β 2 are the roots of the equation 8 x 2 − 26 x + 15 = 0 , then cos ⁡ ( α + β ) =

If 2 sin ⁡ A 2 = 1 + sin ⁡ A + 1 − sin ⁡ A , then A 2 lies between

If the angles of a triangle are 30° and 45° and the included side is ( 3 + 1 ) ems, then the area of the triangle, is

If sin ⁡ ( π cot ⁡ θ ) = cos ⁡ ( π tan ⁡ θ ) then

1 + sin ⁡ x + sin 2 ⁡ x + … to ∞ = 4 + 2 3 If

The value of sin 2 ⁡ 5 ∘ + sin 2 ⁡ 10 ∘ + sin 2 ⁡ 15 ∘ + … + sin 2 ⁡ 85 ∘ + sin 2 ⁡ 90 ∘ is

If sin ⁡ B = 1 5 sin ⁡ ( 2 A + B ) , then tan ⁡ ( A + B ) tan ⁡ A is equal to

sin ⁡ 7 θ + 6 sin ⁡ 5 θ + 17 sin ⁡ 3 θ + 12 sin ⁡ θ sin ⁡ 6 θ + 5 sin ⁡ 4 θ + 12 sin ⁡ 2 θ is equal to

If tan 3 A tan A = k , then sin 3 A sin A is equal to

The minimum value of cos ⁡ 2 θ + cos ⁡ θ for real values of θ , is

If sin ⁡ θ + cos ⁡ θ = 2 cos ⁡ θ then cos ⁡ θ − sin ⁡ θ is equal to

If x cos ⁡ θ = y cos ⁡ θ − 2 π 3 = z cos ⁡ θ + 2 π 3 , then x + y + z =

If tan ⁡ θ = − 4 / 3 , then sin θ is

If ( sec ⁡ A − tan ⁡ A ) ( sec ⁡ B − tan ⁡ B ) ( sec ⁡ C − tan ⁡ C ) = ( sec ⁡ A + tan ⁡ A ) ( sec ⁡ B + tan ⁡ B ) ( sec ⁡ C + tan ⁡ C ) then each side is equal to

In a △ A B C , a = 13 cm , b = 12 cm and c = 5 cm . The distance of A from B C , is

If cos ⁡ ( x − y ) , cos ⁡ x and cos ⁡ ( x + y ) are in H.P., then cos ⁡ x sec ⁡ y 2 equal to

If cos ⁡ ( θ − α ) , cos ⁡ θ , cos ⁡ ( θ + α ) are in H.P., then cos ⁡ θ sec ⁡ ( α / 2 ) is equal to

For x ∈ R , tan ⁡ x + 1 2 tan ⁡ x 2 + 1 2 2 tan ⁡ x 2 2 + … + 1 2 n − 1 tan ⁡ x 2 n − 1 is equal to

If π < α < 3 π 2 , the the expression 4 sin 4 ⁡ α + sin 2 ⁡ 2 α + 4 cos 2 ⁡ π 4 − α 2 is equal to

If, in a △ A B C , ( a + b + c ) ( b + c − a ) = λ b c , then

The value of cos 9°-sin 9°, is

If cos ⁡ A + cos ⁡ B + cos ⁡ C = 0 then cos ⁡ 3 A + cos ⁡ 3 B + cos ⁡ 3 C is equal to

The value of tan 20° + 2 tan 50° – tan 70°, is

The value of tan ⁡ 1 ∘ tan ⁡ 2 ∘ tan ⁡ 3 ∘ … tan ⁡ 89 ∘ is

The value of tan 82 1 ∘ 2 , is

If π 2 < θ < π , then 1 − sin ⁡ θ 1 + sin ⁡ θ + 1 + sin ⁡ θ 1 − sin ⁡ θ is equal to is

The value of log tan 1 ∘ + log ⁡ tan ⁡ 2 ∘ + … + log ⁡ tan ⁡ 89 ∘ is

The period of the function f ( x ) = cos ⁡ x 2 + | sin ⁡ x | is kπ , then k is equal to

General solution of trigonometric equation 1 − cos x = sin x

The number of values of θ in 0 , 3 π satisfying sin 3 θ − cos 3 θ + 1 + 3 sin θ cos θ = 0

Number of integral values of ‘a’ satisfying the equation cos x 2 + cos x = 2 a is (where [ ] denotes G.I.F)

The set of values of a for which the equation cos 2 x + a sin x = 3 a − 17 possesses a solution is c , d then d-c =

If the equation tan 4 x − 2 sec 2 x + a 2 = 0 has at least one solution, then sum of all possible integral values of ‘a’ is equal to

Let θ ∈ 0 , 4 π satisfy the equation sin θ + 2 sin θ + 3 sin θ + 4 = 6 .If the sum of all the values of θ is of the form k π , then the value of k is

The number of integral values of k belonging to 0 , 2 ∪ 5 , ∞ such that the equation k cos x − 3 sin x = k + 1 possess a solution is

Total number of solutions for the equation is x 2 − 3 sin x − π 6 = 3 (where [.] denotes the greatest integer function)

Number of values of θ for the equation cos θ + cos 3 θ + cos 5 θ + cos 7 θ = 0 in 0 , π is

The number of solutions of sin 2 x cos 2 x = 1 + cos 2 x sin 4 x in [ 0 , 2 π ] is

If x ≠ n π 2 and cos x sin 2 x − 3 sin x + 2 = 1 then the general solutions of x is

The general solution of the equation 1 + sin x + …. + sin n x + ….. 1 − sin x + …. + − 1 n sin n x + …. = 1 − cos 2 x 1 + cos 2 x , x ≠ 2 n + 1 π / 2 , n ∈ I is

Number of values of θ in 0 , 2 π for which at least two roots of equation x 3 − 1 + tan θ + cot θ x 2 + tan θ cot θ + tan θ + cot θ x − tan θ cot θ = 0 are equal is

For x ∈ 0 , π , the equation sin x + 2 sin 2 x − sin 3 x = 3 has

The number of values of θ in [ 0 , 3 π ] satisfying sin 3 θ + cos 3 θ + 3 sin θ cos θ – 1 = 0 is

If x ∈ 0 , π 2 , one solution of 3 − 1 sin x + 3 + 1 cos x = 4 2 is π 12 , t h e o t h e r s o l u t i o n i s λ π 36 where λ =

General solution of the equation sin 4 x + cos 4 x = sin x cos x is

Number of integral values of ‘a’ satisfy the equation sin x 2 + sin x − 2 a = 0 is (where [.] denotes G.I.F)

The set of values of ‘a’ for which the equation cos 2 x + a sin x = 2 a − 7 possesses a solution is c , d , then c + d =

The number of solutions of the equation cos x = x is (where [.] denotes the greatest integer function)

If 15 sin 4 x + 10 cos 4 x = 6 , then tan 2 x =

If sin θ cos 3 θ + sin 3 θ cos 9 θ + sin 9 θ cos 27 θ = A tan B θ − tan C θ , then B + C A = A > 0 , θ ≠ 2 n + 1 π 2 , n ∈ Z

Number of solutions of sin x = cos x in 0 , 2 π is (Here x is fractional part of x)

If f : R S is given by f x = sin x − 3 cos x + 1 is onto, then interval of S is

If sin 2 A = x then sin A sin 2 A sin 3 A sin 4 A is a polynomial in x, the sum of whose coefficients is

If the equation cot 4 x − 2 cos e c 2 x + a 2 = 0 has at least one solution then, sum of all possible integral values of ‘a’ is equal to

If 6 cos 2 θ + 2 cos 2 θ / 2 + 2 sin 2 θ = 0 where − π < 0 < π , then θ is equal to

Number of solutions of the equation sin π x 2 3 = x 2 − 2 3 x + 4 is

Number of solutions of the equation 4 sin 2 x + tan 2 x + cot 2 x + cos e c 2 x = 6 in 0 , π is

Number of values of θ in 0 , 2 π for which at least two roots of equation x 3 − 1 + cos θ + sin θ x 2 + cos θ sin θ + cos θ + sin θ x − sin θ c o s θ = 0 are equal is

If 2 tan 2 x − 5 sec x = 1 for exactly 7 distinct values of x ∈ 0 , n π 2 , n ∈ N then the greatest value of n is

The sum of the roots of the equation 4 cos 3 x − 4 cos 2 x − cos π + x − 1 = 0 in the interval 0 , 315 is p π , where p is equal to

If 1 6 sin θ , cos θ and tan θ are in GP, and range of θ is 0 , n π and has 5 values, then n =

Solution of the equation sin x = 5 − 1 4

General solution of the equation tan x = − 3

General solution of equation 7 cos 2 θ + 3 sin 2 θ = 4 is

The solution of x, which satisfy the equation tan 3 x − tan 2 x 1 + tan 3 x . tan x = 1 is

General solution of simultaneous equations tan θ = − 1 3 and cos θ = 3 2 is

General solution of 3 cos x − sin x = 1 is

The equation a sin x + b cos x = c , where c > a 2 + b 2 has

General solution of tan 2 x + 1 − 3 tan x − 3 = 0 is

No of solutions of equation tan x + s e c x = 2 cos x , x ∈ [ 0 , 2 π ] is

The number of integral values of K for which the equation 7 cos θ + 5 sin θ = 2 K + 1 has at least one solution is

General solutions of the equations sin x = − 1 2 and cos x = − 3 2 is

General solution of the equation 1 + cos 3 θ = 2 cos 2 θ is

General solution of the equation 2 cos 2 θ + 11 sin θ = 7 is

Sum of the solutions in [ 0 , 8 π ] of the equation tan x + cot x + 1 = cos x + π 4 is k π then 2K =

If the general solution of the trigonometric equation tan 3 x + tan 2 x + tan x = tan 3 x . tan 2 x . tan x is n π k then k =

If sin 2 θ 1 + sin 2 θ 2 + sin 2 θ 3 ≥ 0 then which of the following is not the possible value of cos θ 1 + cos θ 2 + cos θ 3 ?

The difference of the maximum and the minimum values of the function 5 cos x + 3 cos ( x + π 3 ) + 8 over R is

The equation sin 4 x + cos 4 x + sin 2 x + α = 0 is solvable for

One root of the equation cos x − x + 1 2 = 0 lies in the interval

The general solution of cos x cos 6 x + 1 = 0 is

The range of y such that the evaluation in x , y + cos x = sin x has a real solution

If 3 sin x + 4 cos a x = 7 has at least one solution then the possible values of a

The sum of all solutions in 0 , 4 π of the equation tan + cot x + 1 = cos x + π 4 is

The smallest positive x satisfying the equation log cos x sin x + log sin x cos x = 2 is

The set of all x in − π 2 , π 2 satisfying 4 sin x − 1 < 5 is given by

Number of integer values of n so that sin x ( sin x + cos x ) = n has at least one solution

Solve sin 2 x + cos 2 y = 2 sec 2 z for x , y and z

log tan x 2 + 4 cos 2 x = 2 , then x =

Solve sin x > − 1 2

If x , y ∈ 0 , 2 π and sin x + sin y = 2 then the value of x + y is

If 3 tan ( θ − 15 ∘ ) = tan ( θ + 15 ° ) , then θ =

Number of integral values of k 7 cos x + 5 sin x = 2 k + 1 for which the equation has solution

The equation cos 8 x + b cos 4 x + 1 = 0 will have a solution if b ∈ ( − ∞ , − k ] , then k + 96 2 =

If cos θ 1 = 2 cos θ 2 and T a n ( θ 1 − θ 2 2 ) T a n ( θ 1 + θ 2 2 ) = − p q , t h e n p + q =

The most general solution of 2 1 + cos x + cos x 2 + cos x 3 + … ∞ = 4 i s 2 n π ± π a , t h e n a =

A vertical pole (more than 50 meters high) consists of two portions. The lower being 1 3 r d of the whole. If the upper portion subtends an angle tan − 1 1 2 at a point in a horizontal plane through the foot of the pole at a distance 40 meters from it, then the height of the pole is

A flagstaff stands at the center of a rectangular field whose diagonal is 1200m, and subtends angles 15 0 and 45 0 at the mid points of the sides of the field. The height of the flagstaff is

The equation 2 cos 2 x 2 sin 2 x = x 2 + 1 x 2 , 0 ≤ x ≤ π 2 has

I f 0 ≤ x ≤ 2 π a n d cos x ≤ sin x , t h e n

T h e m o s t g e n e r a l v a l u e o f θ f o r w h i c h sin θ − cos θ = min a ∈ R 1 , a 2 − 6 a + 11 a r e g i v e n b y

In − π 2 , π 2 , log sin θ cos 2 θ = 2 has

I f α , β a r e s o l u t i o n s o f sin 2 x + a sin x + b = 0 a n d cos 2 x + c cos x + d = 0 , then sin α + β is equal to

I f 3 sin π x + cos π x = x 2 − 2 3 x + 19 9 , t h e n x i s e q u a l t o

I f 1 − tan x 1 + tan x = tan y a n d x − y = π 6 , t h e n x , y a r e r e s p e c t i v e l y

I f n b e t h e n u m b e r o f s o l u t i o n s o f t h e e q u a t i o n cot x = cot x + 1 sin x 0 < x < 2 π , t h e n n =

I f θ ∈ 0 , 5 π a n d r ∈ R s u c h t h a t 2 sin θ = r 4 − 2 r 2 + 3 , t h e n t h e m a x i m u m n u m b e r o f v a l u e s o f t h e p a i r r , θ i s

T h e g e n e r a l s o l u t i o n o f t h e e q u a t i o n 1 − sin x + … + − 1 n sin n x + …… 1 + sin x + … + sin n x + …… = 1 − cos 2 x 1 + cos 2 x i s

The least difference between the roots, in the first quadrant 0 ≤ x ≤ π 2 of the equation 4 cos x 2 − 3 sin 2 x + cos 2 x + 1 = 0 , is

I f 0 ≤ a ≤ 3 , 0 ≤ b ≤ 3 a n d t h e e q u a t i o n x 2 + 4 + 3 cos a x + b = 2 x , h a s a t l e a s t o n e s o l u t i o n , t h e n t h e v a l u e o f a + b

I f 0 ≤ x < 2 π , t h e n t h e n u m b e r o f r e a l v a l u e s o f x , w h i c h s a t i s f y t h e e q u a t i o n cos x + cos 2 x + cos 3 x + cos 4 x = 0 , i s

T h e n u m b e r o f s o l u t i o n s o f ∑ r = 1 5 cos r x = 5 i n t h e i n t e r v a l 0 , 2 π i s

G e n e r a l s o l u t i o n o f 3 − 1 cos θ + 3 + 1 sin θ = 2 i s

I f tan A − B = 1 , sec A + B = 2 3 , t h e n l e a s t p o s i t i v e v a l u e s o f A , B a r e

T h e v a l u e s o f ‘ b ’ s u c h t h a t t h e e q u a t i o n b cos x 2 cos 2 x − 1 = b + sin x cos 2 x − 3 sin 2 x tan x p o s s e s s s o l u t i o n s , b e l o n g t o t h e s e t

The values of ‘b’ such that the equation b cos x 2 cos 2 x − 1 = b + sin x cos 2 x − 3 sin 2 x tan x possess solutions, belong to the set

T h e n u m b e r o f p o i n t s i n t e r s e c t i o n o f t h e t w o c u r v e s y = 2 sin x a n d y = 5 x 2 + 2 x + 3 i s

L e t n b e a p o s i t i v e i n t e g e r s u c h t h a t sin π 2 n + cos π 2 n = n 2 , t h e n n =

A tower subtends angles α , 2 α and 3 α respectively at points A, B and C, all lying on a horizontal line through the foot of the tower. Then AB/BC=

The angles of elevation of the top of a tower from two points distant ‘a’ and ‘b’ from the base and in the same straight line with it are complementary. The height of the tower is

A flag staff stands upon the top of a building. At a distance d meters the angles of elevation of the tops of the flag staff and building are α and β respectively. The height of the flag staff is

The sum of angles of elevation of the top of a tower from two points distant ‘a’ and ‘b’ from the base and in the same straight line with it is 90 0 . Then the height of the tower is

AB is a vertical pole. The end A is on the level ground. ‘C’ is the middle point of AB. P is a point on the level ground. The portion CB subtends an angle β at P. If AP=nAB, then tan β =

The distance of three points on a pole from its foot are in A.P. If the angles of elevation of these points from a point on the ground are α , β and γ , then c o t β c o t γ , c o t γ c o t α a n d c o t α c o t β a r e i n

The angles of elevation of a flying kite at three points A,B,C on a horizontal line (lying in the same vertical plane with the kite) are in the ratio 1 : 2 : 3. If AB=a, BC=b, then the height of the kite is

If each side of length a of an equilateral triangle subtends an angle of 60 0 at the top of a tower h meter high situated at the center of the triangle, then

A tree is leaning towards east. Two points due west of the tree are at distances ‘a’ and ‘b’ from its foot. If α and β are the elevations of the top of the tree from these points, its inclination to the horizon is

2 + sin 6 θ + cos 6 θ sin 2 θ + cos 4 θ =

If g(x) is a periodic function then

If sin θ = a − 4 2 then ‘a’ lies in

In a triangle tan A + tan B + tan C = 6 and tan A tan B = 2 . Then the values of tan A , tan B a n d tan C a r e

I f sin θ 1 + sin θ + cos θ 1 + cos θ = x a n d sin θ 1 − sin θ + cos θ 1 − cos θ = y t h e n

If x = cos e c 2 θ : y = sec 2 θ , z = 1 1 − sin 2 θ cos 2 θ they x y z =

If α , β , γ and δ are the solution of the equation tan θ + π 4 = 3 tan 3 θ , no two of which have equal tangents then the value of tan α + tan β + tan γ + tan δ =

If cos α + cos β = sin α + sin β ,then cos 2 α + cos 2 β =

cos 2 x − 2 cos x = 4 sin x − sin 2 x If:

cos sin x = 1 2 then x must lie in the interval :

If sin 3 x sin 3 x = ∑ r = 0 n a r cos r x is an identity then n + a 1 =

The greatest among sin 1 + cos 1 , sin 1 + cos 1 , sin 1 − cos 1 and 1 is

If a sin 2 x + b cos 2 x = c , b sin 2 y + a cos 2 y = d and a tan x = b tan y t h e n a 2 b 2 =

If A,B,C are angles of a triangle such that A is obtuse then

If a sec θ + b tan θ = c then a tan θ + b sec θ 2 =

x = cot 6 ° cot 42 ° , y = tan 66 ° tan 78 ° then y x =

4 sin 420 ° − α cos 60 ° + α =

Number of solution of sec x cos 5 x + 1 = 0 where 0 < x ≤ π 2

The solution set of equations cos 5 x = 1 + sin 4 x is

Number of solution of the equation sin x = x (Where [.] denotes the greatest integer function) is

If A + B = π 3 where A , B ∈ R + . Then the minimum value of sec A + sec B is

If x = sin θ sin θ , y = cos θ cos θ , w h e r e 99 π 2 ≤ θ ≤ 50 π t h e n

If A = cos cos x + sin cos x . Then least and greatest value of A are:

If 0 < x < π 2 then

If sin θ sin ϕ 2 = tan θ tan ϕ = 3 t h e n which of the following is not true.

For 0 < θ < π 2 , tan θ + tan 2 θ + tan 3 θ = 0 if

If 0 < x < π 2 and sin n x + cos n x > 1 then

The number of real solutions of the equation sin x = 2 x + 2 − x is zero then

Let P = a cos θ − b sin θ Then for all real θ

If a cos x + b cos 3 x ≤ 1 ∀ x ∈ R , t h e n b

The number of values of x ∈ 0 , 4 π satisfying 3 cos x − sin x ≥ 2 is

cos 2 x + a sin x = 2 a − 7 p o s s e s s e s a s o l u t i o n f o r :

The solution set of the equation sin π x 2 3 = x 2 − 2 3 x + 4

The set of all x ∈ − π , π satisfying 4 sin x − 1 < 5 is

If A,B,C are angles of a triangle such that A is obtuse then

sin B − C cos B cos C + sin C − A cos C cos A + sin A − B cos A cos B = λ then λ =

If 1 + tan α 1 + tan 4 α = 2 , α ∈ 0 , π 16 then α =

tan 70 o − tan 20 o 2 tan 50 o − tan 50 o − tan 40 o 2 tan 10 o 2020 =

In an acute angled triangle cot A cot B + cot C cot A + cot B cot C =

If A + B = 225 o then the value of cot A 1 + cot A . cot B 1 + cot B =

If tan A − tan B = 2 tan A − B then A , B can be

If A = 35 0 , B = 15 0 and C = 40 0 then tan A tan B + tan B tan C + tan C tan A =

4 sin 7 o sin 67 o sin 53 o sin 21 o =

tan A , tan B , tan C are the roots of the cubic equation x 3 − 7 x 2 + 11 x − 7 = 0 then A + B + C c a n b e

If A, B, C are in A.P and B = π 4 then tan A tan B tan C =

A, B, C are the angles in a triangle cot A , cot B , cot C are the roots of x 3 − λ 1 x 2 + λ 2 x − λ 3 = 0 then

If A,B,C are the roots of x 3 + λ x + 1 = 0 , then tan A + tan B + tan C =

If A + B + C = π then the value of tan A + tan B + tan C tan A . tan B . tan C + cot A tan B + cot B tan C + cot C tan A

In Δ A B C tan A + tan B + tan C = 6 a n d tan A · tan B = 2 then the possible value of sin 2 A : sin 2 B : sin 2 C =

tan A + tan B + tan A tan B − 1 = 0 then

If sin α sin β − cos α cos β + 1 = 0 then 1 + cot α tan β =

If tan α = 1 + 2 – x – 1 and tan β = 1 + 2 x + 1 – 1 . Then α + β =

1 − cot 1 o 1 − cot 2 o 1 − cot 3 o …………… 1 − cot 44 o = 2 n , t h e n n =

sin 23 π 24 = 2 p − q − 1 4 r then the value of p 2 − q 2 − r 2 =

If a n + 1 = 1 2 1 + a n , then cos 1 – a 0 2 a 1 a 2 … … . ∞ =

If A+B+C=0, then The value of A tan A c o r A cot B B tan B cot B cot C C tan C cot C cot A = λ tan A tan B tan C 1 − A tan B + tan C cot C cot B + cot A C tan C cot C cot A

If A D 1 B E 1 C F 1 = π π 2 3 B − C E − F 0 C F 1 then Statement 1: tan A + tan B + tan C = tan A . tan B . tan C Statement 2: cot D + cot E + cot F = cot D . cot E . cot F Then

The value of cot 7 π 16 + 2 cot 3 π 8 + cot 15 π 16 =

2 cos θ + sin θ = 1 then 7 cos θ + 6 sin θ =

In Δ A B C cos A + cos B + cos C = 7 4 and r R = k 4 , then k =

x = sin | sin θ | y = cos θ | cos θ | where 99 π 2 ≤ θ ≤ 50 π , then

cot 7 1 2 o =

If a sin x + b cos ( x + θ ) + b cos ( x – θ ) = d then the minimum value of | cos θ | =

If a n + 1 = 1 2 1 + a n , then cos 1 – a 0 2 a 1 a 2 … … ∞ =

Let 0 ≤ a , b , c , d ≤ π and a,b,c,d are not complimentary such that 2 cos a + 6 cos b + 7 cos c + 9 cos d = 0 and 2 sin a − 6 sin b + 7 sin c − 9 sin d = 0 . Then the value of cos a + d cos b + c =

Let x = sin 1 0 then the value of 1 cos 0 0 cos 1 0 + 1 cos 1 0 cos 2 0 + … … … … + 1 cos 44 0 cos 45 ° =

If 0 < x < π 2 , then

If A , B , C ∈ – π 2 π 2 , then max value of cos A + cos B + cos C

The maximum value of y = 1 sin 6 x + cos 6 x

The range of the function f ( x ) = 16 sec 2 x + 9 cos e c 2 x

If A + B + C = π , then the minimum value of tan 2 A 2 + tan 2 B 2 + tan 2 C 2

No. of solutions of y = sin ( x + 1 ) , y = sin x in [ – 2 π , 2 π ]

Which of the following is the least

Which of the following is greatest

No. of solutions of sin x = e x + e – x

If sin θ < θ < tan θ , then θ ∈

Let A = sin 8 θ + cos 14 θ ,then for all real θ

If A = tan 1 , B = tan 2 , C = tan 3 Then descending order of A, B, C

f ( x ) = 2 + cos 2 x + sec 2 x its value always

In ∆ A B C , A is obtuse then

let α & β be two real roots of the equation k + 1 tan 2 x − 2 . λ tan x = 1 − k , where k ≠ − 1 and λ are real numbers. If tan 2 α + β = 50 then the value of λ is

The value of cos 3 π 8 · cos 3 π 8 + sin 3 π 8 · sin 3 π 8 is:

The number of distant solutions of the equation, log 1 2 sin x = 2 − log 1 2 cos x in the interval , 0 , 2 π is

The general solution of sin 10 x + cos 10 x = 29 16 cos 4 2 x is

If sin x α + cos x α ≥ 1 , 0 < α < π 2 , then

What is the fundamental period of f ( x ) = sin x + sin 3 x cos x + cos 3 x ?

The period of 5 sin π x 4 + 4 sin π x 3 + cos π x 2

Total number of solutions for the equation x 2 − 3 sin ⁡ x − π 6 = 3 is (where [.] denotes the greatest integer function)

If 9 − 8 cos ⁡ 40 ∘ = a + b sec ⁡ 40 ∘ a , b ∈ I , value of a + b is

If x∈ ( 3 π 4 , π ) , t h e n t h e v a l u e o f c o s x + c o s 2 x ( 2 c o s x + 1 ) ( c o s x – 1 ) i s

In a △ABC, if r=1,R=3,s=5, then the value of a 2 + b 2 + c 2 i s

Number of roots of the equation s i n x c o s x + 2 + t a n 2 x + c o t 2 x = 3 , x ∈ 0 , 4 π i s

If 4 x 2 + 2 cos 2 ⁡ θ = 4 x − sin 2 ⁡ θ then the maximum number of ordered pairs ( x , θ ) in [ 0 , 2 π ] is

The minimum distance of the curve a 2 x 2 + b 2 y 2 = 1 from origin is ( a , b > 0 )

If the angles of elevations of the top of a tower from three collinear points A , B and C on a line leading to the foot of the tower are 30 ∘ , 45 ∘ , and 60 ∘ respectively, then the ratio A B : B C is

If tan ⁡ π − 2 α 4 ⋅ tan ⁡ π − 2 β 4 ⋅ tan ⁡ π − 2 γ 4 = 1 , then the value of 212 ( sin ⁡ α ⋅ sin ⁡ β ⋅ sin ⁡ γ + sin ⁡ α + sin ⁡ β + sin ⁡ γ ) is

ABC is a triangular park with AB = AC = 100 m . A clock tower is situated at the midpoint of BC . The angles of elevation of the top of the tower at A and B are cot − 1 ⁡ 3.2 , cosec − 1 ⁡ 2.6 The height of tower is … mt

If tan ⁡ θ = 1 2 + 1 2 + 1 2 + … ∞ , where θ ∈ ( 0 , 2 π ) , then the number of possible values of θ is

Let f ( x + y ) = f ( x ) f ( y ) for all x and y and f ( 1 ) = 2 . If in a triangle ABC , a = f ( 3 ) , b = f(1) + f(3), c = f(2) + f(3) then 2A is equal to

A person standing on the bank of river observes that the angle of elevation of the top of a tree on the opposite bank of the river is 60 0 and when he retires 40 meters away from the tree the angle of elevation becomes 30 0 . Then the breadth of the river is

In a △ ABC , a = 6 , b = 3 and cos ⁡ C = 3 2 , then the area of triangle =

If α , β are complementary angles, sin α = 3 5 ; then sin α cos β − cos α sin β =

Let a 1 = cos ⁡ ( sin ⁡ πx ) , a 2 = sin ⁡ ( cos ⁡ πx ) and a 3 = cos ⁡ ( π ( x + 1 ) ) , where − 1 2 < x < 0 .What are the relative sizes of a 1 , a 2 and a 3 ?

The number of solutions of ( sin ⁡ 2 x + cos ⁡ 2 x ) 1 + sin ⁡ 4 x = 2 in [ − π , π ] is equal to

Suppose 9 − 8 cos ⁡ 40 ∘ = a + bsec ⁡ 40 ∘ ,where a and b are rational numbers. Then a + b equals.

If the minimum value of ∣ sin ⁡ x + cos ⁡ x + tan ⁡ x + cot ⁡ x + sec ⁡ x + cosec ⁡ x ∣ , x ∈ R is m + n then m + n equal ( m , n ∈ z )

Number of values of the parameter α ∈ [ 0 , 2 π ] for which the quadratic function, ( sin ⁡ α ) x 2 + 2 cos ⁡ αx + 1 2 ( cos ⁡ α + sin ⁡ α ) is the square of a linear function is

If T a n θ + S e c θ = 3 , then the principal value of θ + π 6 is

If 3 T a n 4 α − 10 T a n 2 α + 3 = 0 then principal values of ‘ α ’ are

Number of solutions of the equation tan x + sec x = 2 cos x in the interval [ 0 , 2 π ] is

The equation 4Sin 2 x + 4Sinx + a 2 – 3 = 0 has a solution if

The smallest positive x satisfying log C o s x S i n x + log S i n x C o s x = 2 is

If y + C o s θ = S i n θ has a real solution then

Number of solutions of sin x = x 10

If cos ⁡ x + cos ⁡ y + cos ⁡ α = 0 and sin ⁡ x + sin ⁡ y + sin ⁡ α = 0 then cot ⁡ x + y 2 is equal to

tan ⁡ 81 ∘ − tan ⁡ 63 ∘ − tan ⁡ 27 ∘ + tan ⁡ 9 ∘ equals

The value of 1 + cos ⁡ 56 ∘ + cos ⁡ 58 ∘ − cos ⁡ 66 ∘ is equal to

If α , β , γ ∈ 0 , π 2 then the value of sin ⁡ ( α + β + γ ) sin ⁡ α + sin ⁡ β + sin ⁡ γ is

The value of sin ⁡ 12 ∘ sin ⁡ 48 ∘ sin ⁡ 54 ∘ is equal to

If sin ⁡ x sin ⁡ y = 1 2 , cos ⁡ x cos ⁡ y = 3 2 , where x , y ∈ 0 , π 2 , then the value of tan (x+y) is equal to

The value of cos 2 ⁡ A 3 − 4 cos 2 ⁡ A 2 + sin 2 ⁡ A 3 − 4 sin 2 ⁡ A 2 is equal to

The maximum value of the expression 1 sin 2 ⁡ θ + 3 sin ⁡ θcos ⁡ θ + 5 cos 2 ⁡ θ is

Given that, f ( nθ ) = 2 sin ⁡ 2 θ cos ⁡ 2 θ − cos ⁡ 3 nθ ,and f ( θ ) + f ( 2 θ ) + f ( 3 θ ) + … + f ( nθ ) = sin ⁡ λθ sin ⁡ θsin ⁡ μθ , then the value of μ − λ , is .

If a = cos 2 and b = sin 7, then

The value of sin 8 ⁡ θ + cos 8 ⁡ θ + sin 6 ⁡ θcos 2 ⁡ θ + 3 sin 4 ⁡ θcos 2 ⁡ θ + cos 6 ⁡ θsin 2 ⁡ θ + 3 sin 2 ⁡ θcos 4 ⁡ θ is equal to

If xsin ⁡ θ = ysin ⁡ θ + 2 π 3 = zsin ⁡ θ + 4 π 3 then

The maximum value of 1 + sin ⁡ π 4 + θ + 2 cos ⁡ π 4 − θ for real values of θ , is

The ratio of the greatest value of 2 − cos ⁡ x + sin 2 ⁡ x is to its least value, is

If x cos α + y sinα = x cos β + y sinβ = 2 a , then cosα cosβ is

If cos ⁡ ( A − B ) = 3 5 and tan ⁡ Atan ⁡ B = 2 then

If A = sin ⁡ 45 ∘ + cos ⁡ 45 ∘ and B = sin ⁡ 44 ∘ + cos ⁡ 44 ∘ then

If cot ⁡ ( α + β ) = 0 , then sin ⁡ ( α + 2 β ) can be

If cot 2 ⁡ x = cot ⁡ ( x − y ) cot ⁡ ( x − z ) ,then cot2x is equal to (where x ≠ ± π / 4 )

If α = π 14 then the value of ( tan ⁡ αtan ⁡ 2 α + tan ⁡ 2 αtan ⁡ 4 α + tan ⁡ 4 αtan ⁡ α ) is

If cos ⁡ x a = cos ⁡ ( x + θ ) b = cos ⁡ ( x + 2 θ ) c = cos ⁡ ( x + 3 θ ) d ,then a + c b + d is equal to

2 − sin ⁡ α − cos ⁡ α sin ⁡ α − cos ⁡ α is equal to

If sin ⁡ ( y + z − x ) , sin ⁡ ( z + x − y ) , sin ⁡ ( x + y − z ) are in A.P., then tan x, tan y, tan z are in

In a triangle ABC , ∠ C = π 2 . If tan ⁡ A 2 and tan ⁡ B 2 are the roots of the equation ax 2 + bx + c = 0 ( a ≠ 0 ) ,then the value of a + b c (where a, b, c are sides of ∆ opposite to angles A, B, C, respectively) is

Let 0 ≤ a , b , c , d ≤ π where b and c are not complementary such that 2 cos a + 6cos b + 7 cos c + 9 cos d = 0 and 2 sin a -6 sin b + 7 sin c – 9 sin d= 0, then the value of 3 cos ⁡ ( a + d ) cos ⁡ ( b + c ) is

The absolute value of the expression tan ⁡ π 16 + tan ⁡ 5 π 16 + tan ⁡ 9 π 16 + tan ⁡ 13 π 16 is

The maximum value of cos 2 ⁡ 45 ∘ + x + ( sin ⁡ x − cos ⁡ x ) 2 is

Number of solutions of the equation 4 cos 2 ⁡ 2 x + cos ⁡ 2 x + 1 + tan ⁡ x ( tan ⁡ x − 2 3 ) = 0 in [ 0 , 2 π ] is

The number of solutions of 12 cos 3 ⁡ x − 7 cos 2 ⁡ x + 4 cos ⁡ x = 9 is

General solution of tan ⁡ θ + tan ⁡ 4 θ + tan ⁡ 7 θ = tan ⁡ θtan ⁡ 4 θ tan 7 θ is

sin 3 ⁡ θ − cos 3 ⁡ θ sin ⁡ θ − cos ⁡ θ − cos ⁡ θ 1 + cot 2 ⁡ θ − 2 tan ⁡ θcot ⁡ θ = − 1 if

The smallest positive value of x (in radians) satisfying the equation log cos ⁡ x ⁡ 3 2 sin ⁡ x = 2 − log sec ⁡ x ⁡ ( tan ⁡ x ) , is

If 2 sin 2 ⁡ π 2 cos 2 ⁡ x = 1 − cos ⁡ ( πsin ⁡ 2 x ) , x ≠ ( 2 n + 1 ) π 2 , n ∈ I then cos 2x is equal to

Let k be sum of all x in the interval [ 0 , 2 π ] such that 3 cot 2 ⁡ x + 8 cot ⁡ x + 3 = 0 then the value of k π is

If cos ⁡ 4 x = a 0 + a 1 cos 2 ⁡ x + a 2 cos 4 ⁡ x s true for all values of x ∈ R then the value of 5 a 0 + a 1 + a 2 is

Number of solutions of the equation sin ⁡ x cos ⁡ 3 x + sin ⁡ 3 x cos ⁡ 9 x + sin ⁡ 9 x cos ⁡ 27 x = 0 in the interval 0 , π 4 is

The value of k if the equation 2 cos x + cos 2kx = 3 has only one solution is

If tan ⁡ 3 θ + tan ⁡ θ = 2 tan ⁡ 2 θ , then θ is equal to n ∈ Z

If A, B, C are the angles of a triangle such that cot ⁡ A 2 = 3 tan ⁡ C 2 , then sin A, sin B, sin C are in

The number of real roots of the equation cosec ⁡ θ + sec ⁡ θ − 15 = 0 lying in [ 0 , π ] is

The number of solutions of the equation sin 3 ⁡ x cos ⁡ x + sin 2 ⁡ x cos 2 ⁡ x + sin ⁡ x cos 3 ⁡ x = 1 in the interval [0,2 π ] is/ are

Number of solutions of the equation ( 3 + 1 ) 2 x + ( 3 − 1 ) 2 x = 2 3 x .

If sin ⁡ θ + cosec ⁡ θ = 2 , then cos 2015 ⁡ θ + cosec 2015 ⁡ θ is equal to

If 0 < θ < π 2 then tan ⁡ θ + sec ⁡ θ − 1 tan ⁡ θ − sec ⁡ θ + 1 is equal to

If A and B be acute positive angles satisfying 3 sin 2 ⁡ A + 2 sin 2 ⁡ B = 1 , 3 sin ⁡ 2 A − 2 sin ⁡ 2 B = 0 then

Number of values of x lying in the interval [ 0 , 4 π ] and satisfying the equation tan ⁡ 5 x + cot ⁡ 3 x = 0 is

cos ⁡ 11 ∘ − sin ⁡ 11 ∘ cos ⁡ 11 ∘ + sin ⁡ 11 ∘ is equal to

If ( 1 − sin ⁡ A ) ( 1 − sin ⁡ B ) ( 1 − sin ⁡ C ) = ( 1 + sin ⁡ A ) ( 1 + sin ⁡ B ) ( 1 + sin ⁡ C ) , then each sides is equal to

2 cos ⁡ π 13 cos ⁡ 9 π 13 + cos ⁡ 3 π 13 + cos ⁡ 5 π 13

Number of solutions of the equation sin ⁡ π x 2 3 = x 2 − 2 3 x + 4 is

Suppose α , β > 0 and α + 2 β = π / 2 , then tan ⁡ ( α + β ) − 2 tan ⁡ α − tan ⁡ β is equal to

If x cos â¡ Î¸ = y cos â¡ ( Î¸ + 2 Ï / 3 ) = z cos â¡ ( Î¸ + 4 Ï / 3 ) then x y + y z + z x =

If sin A = sin B and cos A = cos B ; A â B , then

6 tan 2 â¡ x â 2 cos 2 â¡ x = cos â¡ 2 x if

If 0 < x < π / 2 and sin ⁡ ( 2 sin ⁡ x ) = cos ⁡ ( 2 cos ⁡ x ) then tan ⁡ x + cot ⁡ x = a π C − b where a + b + c =

If cosec ⁡ A + sec ⁡ A = cosec ⁡ B + sec ⁡ B then tan ⁡ A tan ⁡ B is equal to

A value of θ lying between θ = 0 and θ = π / 2 and satisfying the equation 1 + sin 2 ⁡ θ cos 2 ⁡ θ 4 sin ⁡ 4 θ sin 2 ⁡ θ 1 + cos 2 ⁡ θ 4 sin ⁡ 4 θ sin 2 ⁡ θ cos 2 ⁡ θ 1 + 4 sin ⁡ 4 θ = 0 is

If tan ⁡ A − tan ⁡ B = x and cot ⁡ B − cot ⁡ A = y , then the value of cot ⁡ ( A − B ) is

If sin ⁡ A , cos ⁡ A and tan ⁡ A are in G.P., then cot 6 ⁡ A − cot 2 ⁡ A =

In a Δ P Q R , if 3 sin ⁡ P + 4 cos ⁡ Q = 16 and 4 sin ⁡ Q + 3 cos ⁡ P = 1 , Then the angle R is equal to

tan ⁡ x + 1 2 tan ⁡ x 2 + 1 2 2 tan ⁡ x 2 2 + … + 1 2 n − 1 tan ⁡ x 2 n − 1 is equal to

x = ∑ n = 0 ∞ cos 2 n ⁡ θ , y = ∑ n = 0 ∞ sin 2 n ⁡ θ , z = ∑ n = 0 ∞ cos 2 n ⁡ θ sin 2 n ⁡ θ | cos ⁡ θ | < 1 , | sin ⁡ θ | < 1 then x + y + z is equal to

The value of log 3 ⁡ tan ⁡ 1 ∘ + log 3 ⁡ tan ⁡ 2 ∘ + … + log 3 ⁡ tan ⁡ 89 ∘ is

Let f ( θ ) = cot ⁡ θ 1 − tan ⁡ θ + tan ⁡ θ 1 − cot ⁡ θ , π < θ < 3 π 4 , then f 9 π 8 is equal to

If 0 < x , y < 2 π , the number of solutions of the system of equations sin ⁡ x sin ⁡ y = 3 / 4 and cos ⁡ x cos ⁡ y = 1 / 4 is

The number of solutions of the equation cos ⁡ ( π x − 4 ) cos ⁡ ( π x ) = 1 is

If A , B , C are the angles of a triangle such that angle A is obtuse then

If cos ⁡ ( x − y ) = a cos ⁡ ( x + y ) , then cot ⁡ x cot ⁡ y is equal to

The solution set of the equation tan ⁡ ( π tan ⁡ x ) = cot ⁡ ( π cot ⁡ x ) is

Sum of the root of the equation 2 sin 2 ⁡ θ + sin 2 ⁡ 2 θ = 2 ; 0 ≤ θ ≤ π / 2 is

tan ⁡ 15 ∘ + tan ⁡ 75 ∘ is equal to

The number of solutions of the equation tan ⁡ x + sec ⁡ x = 2 cos ⁡ x , x ∈ [ 0 , 2 π ] is

The numbers of solutions of the pair of equations 2 sin 2 ⁡ θ − cos ⁡ 2 θ = 0 , 2 cos 2 ⁡ θ − 3 sin ⁡ θ = 0 in the interval [ 0 , 2 π ] is

The values of x between 0 and 2 π which satisfy the equation sin ⁡ x 8 cos 2 ⁡ x = 1 are in A.P. The common difference of the A.P. is

The number of solutions of sin ⁡ θ + 2 sin ⁡ 2 θ + 3 sin ⁡ 3 θ + 4 sin ⁡ 4 θ = 10 , 0 < θ < π is

If x y = cos ⁡ A cos ⁡ B then x tan ⁡ A + y tan ⁡ B x + y =

If x = sin ⁡ 2 π 7 + sin ⁡ 4 π 7 + sin ⁡ 8 π 7 and y = cos ⁡ 2 π 7 + cos ⁡ 4 π 7 + cos ⁡ 8 π 7 , then x 2 + y 2 is

If the angles A , B , C of a triangle are in A . P . such that sin ⁡ ( 2 A + B ) = 1 / 2 then sin ⁡ ( B + 2 C ) =

If D = 1 cos ⁡ θ 1 − sin ⁡ θ 1 − cos ⁡ θ − 1 sin ⁡ θ 1 then D lies in the interval

15 [ tan ⁡ 2 θ + sin ⁡ 2 θ ] + 8 = 0 if

If a cos ⁡ A − b sin ⁡ A = c , then a sin ⁡ A + b cos ⁡ A is equal to

2 cos 2 â¡ x + 4 cos â¡ x = 3 sin 2 â¡ x if

If sin â¡ ( Î¸ / 2 ) = a , cos â¡ ( Î¸ / 2 ) = b , then ( 1 + sin â¡ Î¸ ) ( 3 sin â¡ Î¸ + 4 cos â¡ Î¸ + 5 ) =

Number of values of x ∈ [ 0 , 5 π ] and satisfying sin 2 ⁡ x = cos 2 ⁡ x is

If tan ⁡ x 2 = tan ⁡ y 3 = tan ⁡ z 5 and x + y + z = π then the value of tan 2 ⁡ x + tan 2 ⁡ y + tan 2 ⁡ z is

cos ⁡ 7.5 ∘ =

If sin ⁡ 5 θ = a sin 5 ⁡ θ + b sin 3 ⁡ θ + c sin ⁡ θ + d , then

If n ∈ I , the line x = n π + π / 2 does not intersect the graph of

If x = a ( cos ⁡ θ + θ sin ⁡ θ ) , y = a ( sin ⁡ θ − θ cos ⁡ θ ) then a θ =

Let θ ∈ ( π / 4 , π / 2 ) , which of the following statements is true?

If n â N and sin â¡ Ï 2 n + cos â¡ Ï 2 n = n 2 then a possible value of n is

If sin â¡ Î± , sin â¡ Î² are the roots of the equation a sin 2 â¡ Î¸ + b sin â¡ Î¸ + c = 0 and sin â¡ Î± + 2 sin â¡ Î² = 1 and a 2 + 2 b 2 + 3 a b + a c =

If sin â¡ Î¸ + cos â¡ Î¸ = a and cos â¡ Î¸ â sin â¡ Î¸ = b , sin â¡ Î¸ ( sin â¡ Î¸ â cos â¡ Î¸ ) + sin 2 â¡ Î¸ sin 2 â¡ Î¸ â cos 2 â¡ Î¸ + sin 3 â¡ Î¸ sin 3 â¡ Î¸ â cos 3 â¡ Î¸ + â¦ is equal to

If cos ⁡ α + cos ⁡ β = a , sin ⁡ α + sin ⁡ β = b and α − β = 2 θ , then cos ⁡ 3 θ cos ⁡ θ =

The value of 3 + cot â¡ 76 â cot â¡ 16 â cot â¡ 76 â + cot â¡ 16 â is

cos â¡ 22 â + cos â¡ 78 â + cos â¡ 80 â =

e sin ⁡ x − e − sin ⁡ x = 4 for

If sin 2 ⁡ ( x + π / 4 ) + 3 cos ⁡ 2 x > 0 , then

If f ( Î¸ ) = sin â¡ Î¸ ( sin â¡ Î¸ + sin â¡ 3 Î¸ ) , then f ( Î¸ )

cos ⁡ 11 ∘ − cos ⁡ 2 ∘ is

Let f ( θ ) = s i n ⁡ θ s i n ⁡ 3 θ s i n ⁡ 5 θ , then f ( π / 14 ) is equal to

If A > 0 , B > 0 and A + B = Ï / 3 then the maximum value of tan A tan B is

If tan ⁡ θ + tan ⁡ ϕ = a , cot ⁡ θ + cot ⁡ ϕ = b , θ − ϕ = α ( ≠ 0 ) then

The equation tan 2 ⁡ x + cot 2 ⁡ x = 4 a cosec 2 ⁡ 2 x has a real solution if

The sum of the roots of the equation 4 cos 3 ⁡ x − 4 cos 2 ⁡ x − cos ⁡ ( π + x ) − 1 = 0 in the in terval [ 0 , 315 ] is p π where p is equal to

Let f ( θ ) = tan 3 ⁡ θ 1 + tan 2 ⁡ θ − cot 3 ⁡ θ 1 + cot 2 ⁡ θ , 0 < θ < π 4 Then f ( θ ) is equal to

The greatest value of cos Î¸ for which cos 5 Î¸ = 0 is

If 2 tan ⁡ α + cot ⁡ β = tan ⁡ β , then the value of tan ⁡ ( β − α ) is

If 2 cos ⁡ θ + sin ⁡ θ = 1 , ( θ ≠ ( 4 k + 1 ) π / 2 k ∈ I ) then 7 cos ⁡ θ + 6 sin ⁡ θ is equal to

cot ⁡ θ − cot ⁡ 3 θ is equal to

The least positive root of the function sin ⁡ x − π / 2 + 1 = 0 lies in the interval

sin 2 ⁡ A + sin 2 ⁡ ( A − B ) + 2 sin ⁡ A cos ⁡ B sin ⁡ ( B − A ) is equal to

If tan ⁡ p θ = tan ⁡ q θ then the values of θ form an A . P . with common difference

If 15 sin 4 ⁡ x + 10 cos 4 ⁡ x = 6 , Then t an 2 ⁡ x =

Sum of integral values of n such that sin x ( 2 sin x + cos x ) = n has at least one real solution is

The number of real solutions of the equation sin ⁡ e x = 2 x + 2 − x is

If tan ⁡ ( θ / 2 ) = cosec ⁡ θ − sin ⁡ θ , then c os 2 ⁡ ( θ / 2 ) is equal to

The general solution of the trigonometrical equation sin ⁡ x + cos ⁡ x = 1 is

sin x , sin 2 x , sin 3 x are in A.P. if

If tan ⁡ x / 2 = cosec ⁡ x − sin ⁡ x , then sec 2 ⁡ ( x / 2 ) =

cos 2 ⁡ u + cos 2 ⁡ ( u + x ) − 2 cos ⁡ u cos ⁡ x cos ⁡ ( u + x ) = 1 / 2 if

The value of cos 4 ⁡ π 8 + cos 4 ⁡ 3 π 8 + cos 4 ⁡ 5 π 8 + cos 4 ⁡ 7 π 8 is

cos ⁡ 3 θ cos 3 ⁡ θ + sin ⁡ 3 θ sin 3 ⁡ θ is equal to

For 0 < θ < π / 4 , sec ⁡ ( 8 θ ) − 1 sec ⁡ ( 4 θ ) − 1 ⋅ tan ⁡ ( 2 θ ) tan ⁡ ( 8 θ ) is equal to

Which of the following gives the least value of A

If 4 n Î± = Ï then the value of tan â¡ Î± tan â¡ 2 Î± tan â¡ 3 Î± â¦ tan â¡ ( 2 n â 1 ) Î± is

Number of values of x ∈ [ 0 , 4 π ] and satisfying the equation sin ⁡ x + cos ⁡ x = 3 / 2 is

The possible values of θ ∈ ( 0 , π ) such that sin ⁡ ( θ ) + sin ⁡ ( 4 θ ) + sin ⁡ ( 7 θ ) = 0 are

If 2 tan â¡ ( Ï / 3 ) cos â¡ ( 2 Ï x ) = 3 , the general solution of the equation is

If tan A , tan B , tan C satisfy the equation 3 tan 3 ⁡ θ − 4 tan 2 ⁡ θ + 3 tan ⁡ θ + 1 = 0 , then A + B + C =

The value of cos ⁡ 2 π 7 + cos ⁡ 4 π 7 + cos ⁡ 6 π 7 + cos ⁡ 7 π 7 is

( 1 + cos ⁡ ( π / 8 ) ) ( 1 + cos ⁡ ( 3 π / 8 ) ) ( 1 + cos ⁡ ( 5 π / 8 ) ) ( 1 + cos ⁡ ( 7 π / 8 ) ) =

If tan â¡ Î¸ , 2 tan â¡ Î¸ + 2 , 3 tan â¡ Î¸ + 3 are in G.P., then the value of 7 â 5 cot â¡ Î¸ 9 â 4 sec 2 â¡ Î¸ â 1 is

The number of values of satisfying the condition sin x + sin 5 x = sin 3 x in the interval [ 0 , π ] is

If ( 1 + 1 + x ) tan â¡ Î± = ( 1 â 1 â x ) then x =

The number of values of sin x satisfying sin 5 x = 5 sin x is

If − π 2 < θ < π 2 , then the minimum value of cos 3 ⁡ θ + sec 3 ⁡ θ is

The value of sin π 18 sin ⁡ 5 π 18 sin ⁡ 7 π 18 , is

If 2 cos ⁡ A 2 = 1 + sin ⁡ A + 1 − sin ⁡ A , then A 2

If sin ⁡ ( α + β ) = 1 , sin ⁡ ( α − β ) = 1 / 2 ; α , β ∈ [ π / 2 ] then tan ⁡ ( α + 2 β ) tan ⁡ ( 2 α + β ) is equal to

sec ⁡ θ = a 2 + b 2 a 2 − b 2 , where a , b , ∈ R , gives real values of θ if and only if

The value of cos ⁡ 1 ∘ cos ⁡ 2 ∘ cos ⁡ 3 ∘ … cos ⁡ 179 ∘ is

In a Δ P Q R , ∠ R = π 2 . If tan ⁡ P 2 and tan ⁡ Q 2 are the roots of equation a x 2 + b x + c = 0 ( a ≠ 0 ) , then

If π < θ < 2 π , then 1 + cos ⁡ θ 1 − cos ⁡ θ is equal to

sin 2 ⁡ θ = ( x + y ) 2 4 x y , where x , y ∈ R , gives real θ if and only if

The acute angle of a rhombus whose side is a mean proportion al between its diagonals is

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