🔷 1. Basic Trigonometric Ratios
For a right-angled triangle:
- sin θ = Perpendicular / Hypotenuse
- cos θ = Base / Hypotenuse
- tan θ = Perpendicular / Base
- cosec θ = Hypotenuse / Perpendicular
- sec θ = Hypotenuse / Base
- cot θ = Base / Perpendicular
🔷 2. Fundamental Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
🔷 3. Reciprocal Identities
- sin θ = 1 / cosec θ
- cos θ = 1 / sec θ
- tan θ = 1 / cot θ
🔷 4. Ratio Relations
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
🔷 5. Trigonometric Values (Standard Angles)
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
🔷 6. Complementary Angles
- 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
🔷 7. Negative Angle Formulas
- sin(−θ) = −sin θ
- cos(−θ) = cos θ
- tan(−θ) = −tan θ
- cot(−θ) = −cot θ
- sec(−θ) = sec θ
- cosec(−θ) = −cosec θ
🔷 8. Sum and Difference Formulas
- 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)
🔷 9. Double Angle Formulas
- sin 2A = 2 sin A cos A
- cos 2A = cos²A − sin²A
= 1 − 2 sin²A
= 2 cos²A − 1 - tan 2A = 2 tan A / (1 − tan²A)
🔷 10. Triple Angle Formulas (Important)
- sin 3A = 3 sin A − 4 sin³A
- cos 3A = 4 cos³A − 3 cos A
- tan 3A = (3 tan A − tan³A) / (1 − 3 tan²A)
🔷 11. Product to Sum Formulas
- sin A sin B = 1/2 [cos(A − B) − cos(A + B)]
- cos A cos B = 1/2 [cos(A − B) + cos(A + B)]
- sin A cos B = 1/2 [sin(A + B) + sin(A − B)]
🔷 12. Sum to Product Formulas
- 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]
🔷 13. Important Identities for Solving Problems
- sin²A = (1 − cos 2A)/2
- cos²A = (1 + cos 2A)/2
- tan²A = (1 − cos 2A)/(1 + cos 2A)