Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
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.