Courses
Q.
List all trigonometry formulas for Class 10
see full answer
High-Paying Jobs That Even AI Can’t Replace — Through JEE/NEET
(Unlock A.I Detailed Solution for FREE)
Best Courses for You
JEE
NEET
Foundation JEE
Foundation NEET
CBSE
Detailed Solution
Trigonometry is the branch of mathematics that studies relationships between angles and sides of triangles. This comprehensive guide presents all essential trigonometry formulas systematically, suitable for Class 10, Class 11, and college-level students.
1. Basic Trigonometric Ratios
These fundamental ratios form the foundation of trigonometry.
| Function | Formula | Definition |
| Sine | sin θ = Opposite/Hypotenuse | Ratio of opposite side to hypotenuse |
| Cosine | cos θ = Adjacent/Hypotenuse | Ratio of adjacent side to hypotenuse |
| Tangent | tan θ = Opposite/Adjacent | Ratio of opposite side to adjacent side |
| Cosecant | cosec θ = 1/sin θ = Hypotenuse/Opposite | Reciprocal of sine |
| Secant | sec θ = 1/cos θ = Hypotenuse/Adjacent | Reciprocal of cosine |
| Cotangent | cot θ = 1/tan θ = Adjacent/Opposite | Reciprocal of tangent |
2. Reciprocal Identities
| Identity | Formula |
| Sine-Cosecant | sin θ = 1/cosec θ |
| Cosine-Secant | cos θ = 1/sec θ |
| Tangent-Cotangent | tan θ = 1/cot θ |
| Cosecant-Sine | cosec θ = 1/sin θ |
| Secant-Cosine | sec θ = 1/cos θ |
| Cotangent-Tangent | cot θ = 1/tan θ |
3. Quotient Identities
| Identity | Formula |
| Tangent | tan θ = sin θ/cos θ |
| Cotangent | cot θ = cos θ/sin θ |
4. Pythagorean Identities
These identities are derived from the Pythagorean theorem.
| Identity | Formula |
| Primary Identity | sin²θ + cos²θ = 1 |
| Derived Identity 1 | 1 + tan²θ = sec²θ |
| Derived Identity 2 | 1 + cot²θ = cosec²θ |
Alternative Forms:
- sin²θ = 1 - cos²θ
- cos²θ = 1 - sin²θ
- tan²θ = sec²θ - 1
- cot²θ = cosec²θ - 1
5. Trigonometric Values of Standard Angles
| Angle | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ | 0 | ∞ | 0 |
| cot θ | ∞ | √3 | 1 | 1/√3 | 0 | ∞ | 0 | ∞ |
| sec θ | 1 | 2/√3 | √2 | 2 | ∞ | -1 | ∞ | 1 |
| cosec θ | ∞ | 2 | √2 | 2/√3 | 1 | ∞ | -1 | ∞ |
6. Sign Convention in Different Quadrants
| Quadrant | Angle Range | sin θ | cos θ | tan θ | cot θ | sec θ | cosec θ |
| I | 0° to 90° | + | + | + | + | + | + |
| II | 90° to 180° | + | - | - | - | - | + |
| III | 180° to 270° | - | - | + | + | - | - |
| IV | 270° to 360° | - | + | - | - | + | - |
Memory Tip: "All Students Take Calculus" (All positive, Sin positive, Tan positive, Cos positive)
7. Complementary Angle Formulas (Co-function Identities)
| Formula | Equivalent |
| sin(90° - θ) = cos θ | cos(90° - θ) = sin θ |
| tan(90° - θ) = cot θ | cot(90° - θ) = tan θ |
| sec(90° - θ) = cosec θ | cosec(90° - θ) = sec θ |
8. Supplementary Angle Formulas
| Formula | Value |
| sin(180° - θ) | sin θ |
| cos(180° - θ) | -cos θ |
| tan(180° - θ) | -tan θ |
| cot(180° - θ) | -cot θ |
| sec(180° - θ) | -sec θ |
| cosec(180° - θ) | cosec θ |
9. Negative Angle Formulas (Even-Odd Identities)
| Formula | Value | Property |
| sin(-θ) | -sin θ | Odd function |
| cos(-θ) | cos θ | Even function |
| tan(-θ) | -tan θ | Odd function |
| cot(-θ) | -cot θ | Odd function |
| sec(-θ) | sec θ | Even function |
| cosec(-θ) | -cosec θ | Odd function |
10. Sum and Difference Formulas (Compound Angle Formulas)
Addition Formulas:
| Formula | Expansion |
| sin(A + B) | sin A cos B + cos A sin B |
| cos(A + B) | cos A cos B - sin A sin B |
| tan(A + B) | (tan A + tan B)/(1 - tan A tan B) |
Subtraction Formulas:
| Formula | Expansion |
| sin(A - B) | sin A cos B - cos A sin B |
| cos(A - B) | cos A cos B + sin A sin B |
| tan(A - B) | (tan A - tan B)/(1 + tan A tan B) |
11. Double Angle Formulas
| Function | Formulas |
| sin 2θ | 2 sin θ cos θ |
| cos 2θ | cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ |
| tan 2θ | 2 tan θ/(1 - tan²θ) |
| sin²θ | (1 - cos 2θ)/2 |
| cos²θ | (1 + cos 2θ)/2 |
| tan²θ | (1 - cos 2θ)/(1 + cos 2θ) |
12. Triple Angle Formulas
| Formula | Expansion |
| sin 3θ | 3 sin θ - 4 sin³θ |
| cos 3θ | 4 cos³θ - 3 cos θ |
| tan 3θ | (3 tan θ - tan³θ)/(1 - 3 tan²θ) |
13. Half Angle Formulas
| Formula | Expression |
| sin(θ/2) | ±√[(1 - cos θ)/2] |
| cos(θ/2) | ±√[(1 + cos θ)/2] |
| tan(θ/2) | ±√[(1 - cos θ)/(1 + cos θ)] = sin θ/(1 + cos θ) = (1 - cos θ)/sin θ |
Note: The sign depends on the quadrant in which θ/2 lies.
14. Product-to-Sum Formulas
| Product | Sum/Difference |
| sin A sin B | [cos(A - B) - cos(A + B)]/2 |
| cos A cos B | [cos(A - B) + cos(A + B)]/2 |
| sin A cos B | [sin(A + B) + sin(A - B)]/2 |
| cos A sin B | [sin(A + B) - sin(A - B)]/2 |
15. Sum-to-Product Formulas
| Sum/Difference | Product |
| sin A + sin B | 2 sin[(A + B)/2] cos[(A - B)/2] |
| sin A - sin B | 2 cos[(A + B)/2] sin[(A - B)/2] |
| cos A + cos B | 2 cos[(A + B)/2] cos[(A - B)/2] |
| cos A - cos B | -2 sin[(A + B)/2] sin[(A - B)/2] |
16. Inverse Trigonometric Functions
Basic Definitions:
| Function | Domain | Range | Notation |
| sin⁻¹x or arcsin x | [-1, 1] | [-π/2, π/2] or [-90°, 90°] | y = sin⁻¹x means x = sin y |
| cos⁻¹x or arccos x | [-1, 1] | [0, π] or [0°, 180°] | y = cos⁻¹x means x = cos y |
| tan⁻¹x or arctan x | (-∞, ∞) | (-π/2, π/2) or (-90°, 90°) | y = tan⁻¹x means x = tan y |
| cot⁻¹x or arccot x | (-∞, ∞) | (0, π) or (0°, 180°) | y = cot⁻¹x means x = cot y |
| sec⁻¹x or arcsec x | (-∞, -1] ∪ [1, ∞) | [0, π/2) ∪ (π/2, π] | y = sec⁻¹x means x = sec y |
| cosec⁻¹x or arccosec x | (-∞, -1] ∪ [1, ∞) | [-π/2, 0) ∪ (0, π/2] | y = cosec⁻¹x means x = cosec y |
17. Properties of Inverse Trigonometric Functions
Reciprocal Properties:
| Property | Formula |
| sin⁻¹(1/x) | cosec⁻¹x, for x ≥ 1 or x ≤ -1 |
| cos⁻¹(1/x) | sec⁻¹x, for x ≥ 1 or x ≤ -1 |
| tan⁻¹(1/x) | cot⁻¹x, for x > 0 |
| tan⁻¹(1/x) | -π + cot⁻¹x, for x < 0 |
Complementary Properties:
| Property | Formula |
| sin⁻¹x + cos⁻¹x | π/2 |
| tan⁻¹x + cot⁻¹x | π/2 |
| sec⁻¹x + cosec⁻¹x | π/2 |
Negative Argument Properties:
| Property | Formula |
| sin⁻¹(-x) | -sin⁻¹x |
| cos⁻¹(-x) | π - cos⁻¹x |
| tan⁻¹(-x) | -tan⁻¹x |
| cot⁻¹(-x) | π - cot⁻¹x |
| sec⁻¹(-x) | π - sec⁻¹x |
| cosec⁻¹(-x) | -cosec⁻¹x |
18. Sum and Difference of Inverse Functions
| Formula | Expression |
| sin⁻¹x + sin⁻¹y | sin⁻¹[x√(1-y²) + y√(1-x²)] |
| sin⁻¹x - sin⁻¹y | sin⁻¹[x√(1-y²) - y√(1-x²)] |
| cos⁻¹x + cos⁻¹y | cos⁻¹[xy - √(1-x²)√(1-y²)] |
| cos⁻¹x - cos⁻¹y | cos⁻¹[xy + √(1-x²)√(1-y²)] |
| tan⁻¹x + tan⁻¹y | tan⁻¹[(x + y)/(1 - xy)], if xy < 1 |
| tan⁻¹x - tan⁻¹y | tan⁻¹[(x - y)/(1 + xy)] |
| 2tan⁻¹x | tan⁻¹[2x/(1 - x²)] = sin⁻¹[2x/(1 + x²)] = cos⁻¹[(1 - x²)/(1 + x²)] |
19. Composition of Trigonometric and Inverse Functions
| Composition | Result | Condition |
| sin(sin⁻¹x) | x | -1 ≤ x ≤ 1 |
| cos(cos⁻¹x) | x | -1 ≤ x ≤ 1 |
| tan(tan⁻¹x) | x | -∞ < x < ∞ |
| sin⁻¹(sin x) | x | -π/2 ≤ x ≤ π/2 |
| cos⁻¹(cos x) | x | 0 ≤ x ≤ π |
| tan⁻¹(tan x) | x | -π/2 < x < π/2 |
20. Important Conversion Formulas
| Conversion | Formula |
| sin⁻¹x | cos⁻¹(√(1-x²)) = tan⁻¹(x/√(1-x²)) |
| cos⁻¹x | sin⁻¹(√(1-x²)) = tan⁻¹(√(1-x²)/x) |
| tan⁻¹x | sin⁻¹(x/√(1+x²)) = cos⁻¹(1/√(1+x²)) |
21. Sine and Cosine Rules for Any Triangle
Sine Rule:
For any triangle ABC with sides a, b, c opposite to angles A, B, C:
| Formula |
| a/sin A = b/sin B = c/sin C = 2R |
Where R is the circumradius of the triangle.
Cosine Rule:
| Formula | Use |
| a² = b² + c² - 2bc cos A | To find a side when two sides and included angle are known |
| b² = a² + c² - 2ac cos B | To find a side when two sides and included angle are known |
| c² = a² + b² - 2ab cos C | To find a side when two sides and included angle are known |
| cos A = (b² + c² - a²)/(2bc) | To find an angle when all three sides are known |
| cos B = (a² + c² - b²)/(2ac) | To find an angle when all three sides are known |
| cos C = (a² + b² - c²)/(2ab) | To find an angle when all three sides are known |
Projection Rule:
| Formula |
| a = b cos C + c cos B |
| b = c cos A + a cos C |
| c = a cos B + b cos A |
22. Area of Triangle Formulas
| Formula Name | Formula | When to Use |
| Using base and height | Area = (1/2) × base × height | When height is known |
| Using two sides and included angle | Area = (1/2)ab sin C = (1/2)bc sin A = (1/2)ac sin B | When two sides and included angle are known |
| Heron's Formula | Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 | When all three sides are known |
| Using circumradius | Area = abc/(4R) | When all sides and circumradius are known |
| Using inradius | Area = rs, where r is inradius and s is semi-perimeter | When inradius and semi-perimeter are known |
23. Maximum and Minimum Values
| Function | Maximum Value | Minimum Value |
| sin θ | 1 (at θ = 90°) | -1 (at θ = 270°) |
| cos θ | 1 (at θ = 0°, 360°) | -1 (at θ = 180°) |
| tan θ | +∞ | -∞ |
| a sin θ + b cos θ | √(a² + b²) | -√(a² + b²) |
| a sin θ + b | a + b | -a + b |
| a cos θ + b | a + b | -a + b |
24. Periodicity of Trigonometric Functions
| Function | Period |
| sin θ | 2π or 360° |
| cos θ | 2π or 360° |
| tan θ | π or 180° |
| cot θ | π or 180° |
| sec θ | 2π or 360° |
| cosec θ | 2π or 360° |
General Rule:
- sin(θ + 2nπ) = sin θ
- cos(θ + 2nπ) = cos θ
- tan(θ + nπ) = tan θ, where n is any integer
25. Allied Angles Formulas
| Angle | sin | cos | tan |
| -θ | -sin θ | cos θ | -tan θ |
| 90° - θ | cos θ | sin θ | cot θ |
| 90° + θ | cos θ | -sin θ | -cot θ |
| 180° - θ | sin θ | -cos θ | -tan θ |
| 180° + θ | -sin θ | -cos θ | tan θ |
| 270° - θ | -cos θ | -sin θ | cot θ |
| 270° + θ | -cos θ | sin θ | -cot θ |
| 360° - θ | -sin θ | cos θ | -tan θ |
courses
No courses found
Ready to Test Your Skills?
Check your Performance Today with our Free Mock Test used by Toppers!
Take Free Test
