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Solutions

Overview

Step-by-step NCERT solutions for Trigonometric Functions (Chapter 3, NCERT Class 11 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Trigonometric Functions textbook chapter.

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What these solutions cover

All 52 questions in Trigonometric Functions are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 3.1

  1. Find the radian measures corresponding to the following degree measures:
    • (i) 25°
    • (ii) –47°30′
    • (iii) 240°
    • (iv) 520°
  2. Find the degree measures corresponding to the following radian measures (Use π = 22/7):
    • (i) 11/16
    • (ii) –4
    • (iii) 5π/3
    • (iv) 7π/6
  3. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
  4. Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm (Use π = 22/7).
  5. In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
  6. If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.
  7. Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length:
    • (i) 10 cm
    • (ii) 15 cm
    • (iii) 21 cm

Exercise 3.2

  1. Find the values of other five trigonometric functions if cos x = –1/2, x lies in third quadrant.
  2. Find the values of other five trigonometric functions if sin x = 3/5, x lies in second quadrant.
  3. Find the values of other five trigonometric functions if cot x = 3/4, x lies in third quadrant.
  4. Find the values of other five trigonometric functions if sec x = 13/5, x lies in fourth quadrant.
  5. Find the values of other five trigonometric functions if tan x = –5/12, x lies in second quadrant.
  6. Find the value of sin 765°.
  7. Find the value of cosec(–1410°).
  8. Find the value of tan(19π/3).
  9. Find the value of sin(–11π/3).
  10. Find the value of cot(–15π/4).

Exercise 3.3

  1. Prove that: sin²(π/6) + cos²(π/3) – tan²(π/4) = –1/2.
  2. Prove that: 2sin²(π/6) + cosec²(7π/6) · cos²(π/3) = 3/2.
  3. Prove that: cot²(π/6) + cosec(5π/6) + 3tan²(π/6) = 6.
  4. Prove that: 2sin²(3π/4) + 2cos²(π/4) + 2sec²(π/3) = 10.
  5. Find the value of:
    • (i) sin 75°
    • (ii) tan 15°
  6. Prove that: cos(π/4 – x)cos(π/4 – y) – sin(π/4 – x)sin(π/4 – y) = sin(x + y).
  7. Prove that: tan(π/4 + x) / tan(π/4 – x) = ((1 + tan x)/(1 – tan x))².
  8. Prove that: [cos(π + x) · cos(–x)] / [sin(π – x) · cos(π/2 + x)] = cot²x.
  9. Prove that: cos(3π/2 + x) · cos(2π + x) · [cot(3π/2 – x) + cot(2π + x)] = 1.
  10. Prove that: sin(n+1)x · sin(n+2)x + cos(n+1)x · cos(n+2)x = cos x.
  11. Prove that: cos(3π/4 + x) – cos(3π/4 – x) = –√2 sin x.
  12. Prove that: sin²6x – sin²4x = sin 2x · sin 10x.
  13. Prove that: cos²2x – cos²6x = sin 4x · sin 8x.
  14. Prove that: sin 2x + 2 sin 4x + sin 6x = 4 cos²x · sin 4x.
  15. Prove that: cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x).
  16. Prove that: (cos 9x – cos 5x)/(sin 17x – sin 3x) = –sin 2x/cos 10x.
  17. Prove that: (sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x.
  18. Prove that: (sin x – sin y)/(cos x + cos y) = tan((x – y)/2).
  19. Prove that: (sin x + sin 3x)/(cos x + cos 3x) = tan 2x.
  20. Prove that: (sin x – sin 3x)/(sin²x – cos²x) = 2 sin x.
  21. Prove that: (cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x) = cot 3x.
  22. Prove that: cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1.
  23. Prove that: tan 4x = 4tan x(1 – tan²x)/(1 – 6tan²x + tan⁴x).
  24. Prove that: cos 4x = 1 – 8sin²x cos²x.
  25. Prove that: cos 6x = 32cos⁶x – 48cos⁴x + 18cos²x – 1.

Miscellaneous Exercise

  1. Prove that: 2cos(π/13)cos(9π/13) + cos(3π/13) + cos(5π/13) = 0.
  2. Prove that: (sin 3x + sin x)sin x + (cos 3x – cos x)cos x = 0.
  3. Prove that: (cos x + cos y)² + (sin x – sin y)² = 4cos²((x + y)/2).
  4. Prove that: (cos x – cos y)² + (sin x – sin y)² = 4sin²((x – y)/2).
  5. Prove that: sin x + sin 3x + sin 5x + sin 7x = 4cos x cos 2x sin 4x.
  6. Prove that: [(sin 7x + sin 5x) + (sin 9x + sin 3x)] / [(cos 7x + cos 5x) + (cos 9x + cos 3x)] = tan 6x.
  7. Prove that: sin 3x + sin 2x – sin x = 4 sin x cos(x/2) cos(3x/2).
  8. Find sin(x/2), cos(x/2) and tan(x/2) if tan x = –4/3 and x is in quadrant II.
  9. Find sin(x/2), cos(x/2) and tan(x/2) if cos x = –1/3 and x is in quadrant III.
  10. Find sin(x/2), cos(x/2) and tan(x/2) if sin x = 1/4 and x is in quadrant II.
Sample

Worked examples

3 questions from Trigonometric Functions, solved step by step.

Exercise 3.1

Find the radian measures corresponding to the following degree measures: (i) 25° (ii) –47°30′ (iii) 240° (iv) 520°

Use the conversion: Radian measure = (π/180) × Degree measure.

(i) 25° = (π/180) × 25 = 25π/180 = 5π/36 radians.

(ii) –47°30′ = –47.5° (since 30′ = 0.5°) = –(47.5π/180) = –(95π/360) = –19π/72 radians.

(iii) 240° = (π/180) × 240 = 240π/180 = 4π/3 radians.

(iv) 520° = (π/180) × 520 = 520π/180 = 26π/9 radians.

Answer: (i) 5π/36 (ii) –19π/72 (iii) 4π/3 (iv) 26π/9

Exercise 3.2

Find the values of other five trigonometric functions if cos x = –1/2, x lies in third quadrant.

Given: cos x = –1/2, x in third quadrant.

sec x = 1/cos x = –2.

sin²x = 1 – cos²x = 1 – 1/4 = 3/4, so sin x = ±√3/2.

Since x is in the third quadrant, sin x is negative: sin x = –√3/2.

cosec x = 1/sin x = –2/√3 = –2√3/3.

tan x = sin x/cos x = (–√3/2)/(–1/2) = √3.

cot x = 1/tan x = 1/√3 = √3/3.

Answer: sin x = –√3/2, cosec x = –2√3/3, sec x = –2, tan x = √3, cot x = √3/3

Exercise 3.3

Prove that: sin²(π/6) + cos²(π/3) – tan²(π/4) = –1/2.

sin(π/6) = 1/2, so sin²(π/6) = 1/4.

cos(π/3) = 1/2, so cos²(π/3) = 1/4.

tan(π/4) = 1, so tan²(π/4) = 1.

LHS = 1/4 + 1/4 – 1 = 1/2 – 1 = –1/2 = RHS.

Answer: Proved. LHS = 1/4 + 1/4 – 1 = –1/2 = RHS.

All 52 questions in Trigonometric Functions are solved step by step in the Solutions PDF above.

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Read the Trigonometric Functions textbook chapter / PDF, or browse all NCERT Class 11 Mathematics solutions.

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