Quick Summary
Trigonometry at Class 11 level goes well beyond the basic ratios of Classes 9–10. It covers compound angles, multiple angles, sub-multiple angles, product-to-sum and sum-to-product formulas, and inverse trigonometric functions. This chapter is heavily tested in board exams, NEET, and JEE.
Trigonometric Ratios — Quick Reference
For angle θ in a right triangle:
| Ratio | Definition | Reciprocal |
|---|---|---|
| sin θ | opposite / hypotenuse | cosec θ = 1/sin θ |
| cos θ | adjacent / hypotenuse | sec θ = 1/cos θ |
| tan θ | opposite / adjacent | cot θ = 1/tan θ |
Key identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
Standard Angle Values
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ |
Compound Angle Formulas
| Formula | Expression |
|---|---|
| sin(A ± B) | sinA cosB ± cosA sinB |
| cos(A ± B) | cosA cosB ∓ sinA sinB |
| tan(A ± B) | (tanA ± tanB) / (1 ∓ tanA tanB) |
Multiple Angle Formulas
| Formula | Forms |
|---|---|
| sin 2A | 2 sinA cosA |
| cos 2A | cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A |
| tan 2A | 2tanA / (1 − tan²A) |
| sin 3A | 3sinA − 4sin³A |
| cos 3A | 4cos³A − 3cosA |
Sub-Multiple Angle (Half-Angle) Formulas
- sin A = 2 sin(A/2) cos(A/2)
- cos A = 1 − 2sin²(A/2) = 2cos²(A/2) − 1
- tan(A/2) = sinA / (1 + cosA) = (1 − cosA) / sinA
Product-to-Sum Formulas
| Product | Sum/Difference |
|---|---|
| 2 sinA cosB | sin(A+B) + sin(A−B) |
| 2 cosA cosB | cos(A−B) + cos(A+B) |
| 2 sinA sinB | cos(A−B) − cos(A+B) |
Sum-to-Product Formulas
| Sum | Product |
|---|---|
| sinC + sinD | 2 sin((C+D)/2) cos((C−D)/2) |
| sinC − sinD | 2 cos((C+D)/2) sin((C−D)/2) |
| cosC + cosD | 2 cos((C+D)/2) cos((C−D)/2) |
| cosC − cosD | −2 sin((C+D)/2) sin((C−D)/2) |
Inverse Trigonometric Functions
| Function | Domain | Range |
|---|---|---|
| sin⁻¹ x | [−1, 1] | [−π/2, π/2] |
| cos⁻¹ x | [−1, 1] | [0, π] |
| tan⁻¹ x | ℝ | (−π/2, π/2) |
Key identities: sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + cot⁻¹x = π/2
Worked Examples
Example 1: Prove that sin(A+B) · sin(A−B) = sin²A − sin²B. LHS = (sinA cosB + cosA sinB)(sinA cosB − cosA sinB) = sin²A cos²B − cos²A sin²B = sin²A(1−sin²B) − (1−sin²A)sin²B = sin²A − sin²B = RHS ✓
Example 2: Find the value of cos 75°. cos 75° = cos(45° + 30°) = cos45°cos30° − sin45°sin30° = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2) = (√6 − √2)/4
Example 3: Prove that sin20° sin40° sin80° = √3/8. Use product formula repeatedly: sin20° sin40° = ½[cos20° − cos60°] = ½[cos20° − ½] Then multiply by sin80° = cos10°… (standard board proof) = √3/8 ✓
Important Questions
One Mark
- Write the value of sin 75°.
- State the formula for cos 2A in terms of cos A only.
- What is the principal value range of tan⁻¹?
- Express sin 3A in terms of sin A.
Short Answer (3 marks)
- Prove: (sin A + sin 3A) / (cos A + cos 3A) = tan 2A.
- Find sin 18° using the identity for sin 3θ = cos 2θ.
- Express 2 cos 3θ cos 5θ as a sum.
Long Answer (5 marks)
- Prove: sin(A+B+C) = sinA cosB cosC + cosA sinB cosC + cosA cosB sinC − sinA sinB sinC.
- In triangle ABC, prove that a/sinA = b/sinB = c/sinC (Sine Rule). Use it to solve a triangle given two angles and one side.
Common Mistakes
- Mixing up sin(A+B) with sinA + sinB — they are NOT equal.
- Using cos 2A = cos²A − sin²A when only one trig ratio is given — use the appropriate form (2cos²A−1 or 1−2sin²A) to simplify faster.
- Forgetting the negative sign in cosC − cosD = −2 sin((C+D)/2) sin((C−D)/2).
- Inverse trig: applying sin⁻¹(sin x) = x without checking if x is in the principal range.
Board Exam Tips
- Proof questions (proving identities) require starting from the more complex side and working toward the simpler side — never cross-multiply.
- Sum-to-product conversions appear in "prove that" questions as 3-mark or 5-mark items.
- The values of sin 18°, cos 36°, cos 18°, and sin 36° are textbook derivations — practise them.
- NEET/JEE: Multiple angle and compound angle problems appear in almost every paper.