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From the following options, find the value of $cos A sin B + sin A ⋅ cos B,$ if $sin A = 45$ and $cos B = 1213$ .

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35

6365

365

answer is C.

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Detailed Solution

Given that,

sin A = 45

and

cos B = 1213

.
Write

sin A = 45

in terms of the ratio of sides of the right triangle.

sin A = 45

⇒ 4 k 5 k = O p p o s i t e s i d e H y p o t e n u s e

With respect to

A

, form a triangle with opposite side (BC) as 4k and the hypotenuse (AC) as 5k.
Question Image

Find third side

A B

by applying Pythagoras theorem.

(b a s e) 2 + (p e r p e n d i c u l a r) 2 = (h y p o t e n u s e) 2

⇒ A C 2 = B C 2 + A B 2 ⇒ 25 k 2 = 16 k 2 + A B 2 ⇒ A B 2 = 25 k 2 − 16 k 2 ⇒ A B = 3 k

A

Write

cos A

from sides of the triangle.

cos θ = A d j a c e n t s i d e H y p o t e n u s e

⇒ cos A = A B A C ⇒ cos A = 3 k 5 k ⇒ cos A = 35

Write

cos B = 1213

in terms of the ratio of sides of the right triangle.

cos B = 1213

12 k 13 k = A d j a c e n t side H y p o t e n u s e

We know that,

cos θ = A d j a c e n t s i d e H y p o t e n u s e

.
With respect to B, form a triangle with adjacent side (AB) as 12k and the hypotenuse (BC) as 13k.
Question Image

Find third side

A C

by applying Pythagoras theorem.

(b a s e) 2 + (p e r p e n d i c u l a r) 2 = (h y p o t e n u s e) 2

⇒ B C 2 = A C 2 + A B 2 ⇒ 169 k 2 = A C 2 + 144 k 2 ⇒ A C 2 = 169 k 2 − 144 k 2 ⇒ A C 2 = 25 k 2 ⇒ A C = 5 k

Write

sin B

from sides of the triangle.

sin θ = O p p o s i t e s i d e H y p o t e n u s e

⇒ sin B = A C B C ⇒ sin B = 5 k 13 k ⇒ sin B = 513

To find the value of expression put the values of

sin B

and

cos A

Thus,

c o s A s i n B + s i n A c o s B = 6365

.
Hence, option 3) is correct.

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From the following options, find the value of cosAsinB+sinA⋅cosB, if sinA= 4 5 and cosB= 12 13 .

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From the following options, find the value of $cos A sin B + sin A ⋅ cos B,$ if $sin A = 45$ and $cos B = 1213$ .