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Gyaan Maths
CBSE · Class 6-10
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CBSE Maths · Class 10

Chapter 8: Introduction to Trigonometry

Name: ________________________
Date: ______________
Marks: ____ / 10

Trigonometric ratios (sin, cos, tan and reciprocals), values at standard angles, and fundamental identities.

Questions

  1. Evaluate sin⁡60∘cos⁡30∘+sin⁡30∘cos⁡60∘\sin 60^\circ\cos 30^\circ + \sin 30^\circ\cos 60^\circ.
  2. Prove (1−cos⁡2θ)csc⁡2θ=1(1 - \cos^2\theta)\csc^2\theta = 1.
  3. If tan⁡θ=8/15\tan\theta = 8/15, find sin⁡θ+cos⁡θ\sin\theta + \cos\theta.
  4. sin⁡45∘+cos⁡45∘\sin 45^\circ + \cos 45^\circ.
  5. cos⁡θ=1/2\cos\theta = 1/2: find θ\theta.
  6. Prove (sec⁡θ+tan⁡θ)(sec⁡θ−tan⁡θ)=1(\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = 1.
  7. If sin⁡A=3/5\sin A = 3/5, find cos⁡A,tan⁡A\cos A, \tan A.
  8. Prove tan⁡θ1−cot⁡θ+cot⁡θ1−tan⁡θ=1+sec⁡θcsc⁡θ\dfrac{\tan\theta}{1-\cot\theta} + \dfrac{\cot\theta}{1-\tan\theta} = 1 + \sec\theta\csc\theta.
  9. Value of tan⁡260+4cos⁡245+3sec⁡230+5cos⁡290\tan^2 60 + 4\cos^2 45 + 3\sec^2 30 + 5\cos^2 90.
  10. If sin⁡θ+cos⁡θ=2\sin\theta + \cos\theta = \sqrt 2, prove tan⁡θ+cot⁡θ=2\tan\theta + \cot\theta = 2.

Extra Practice (Optional)

  • Evaluate tan⁡45sec⁡60+csc⁡30\tan 45\sec 60 + \csc 30.
  • cos⁡θ=12/13\cos\theta = 12/13: find sin⁡θ\sin\theta.
  • sin⁡(90−θ)cos⁡θ+cos⁡(90−θ)sin⁡θ\sin(90-\theta)\cos\theta + \cos(90-\theta)\sin\theta.
  • Prove 1+cos⁡θsin⁡θ=csc⁡θ+cot⁡θ\dfrac{1+\cos\theta}{\sin\theta} = \csc\theta + \cot\theta.
  • Simplify cos⁡θ1−tan⁡θ+sin⁡θ1−cot⁡θ\dfrac{\cos\theta}{1-\tan\theta} + \dfrac{\sin\theta}{1-\cot\theta}.

Answer Key (For Teachers)

  1. Answer: 11.
    Working: Sum of values gives 3/4+1/43/4 + 1/4.
  2. Answer: Proved.
    Working: sin⁡2θ⋅1/sin⁡2θ\sin^2\theta\cdot 1/\sin^2\theta.
  3. Answer: 23/1723/17.
    Working: Hyp 1717; sum =23/17= 23/17.
  4. Answer: 2\sqrt 2.
    Working: Two halves of 2\sqrt 2.
  5. Answer: 60∘60^\circ.
    Working: Standard table.
  6. Answer: Proved.
    Working: sec⁡2−tan⁡2=1\sec^2 - \tan^2 = 1.
  7. Answer: 4/5,3/44/5, 3/4.
    Working: Right triangle.
  8. Answer: Proved.
    Working: Combine and simplify using sin⁡,cos⁡\sin, \cos forms; standard identity proof.
  9. Answer: 99.
    Working: 3+4(1/2)+3(4/3)+0=3+2+4=93 + 4(1/2) + 3(4/3) + 0 = 3 + 2 + 4 = 9.
  10. Answer: Proved.
    Working: Square: 1+2sin⁡θcos⁡θ=2⇒sin⁡θcos⁡θ=1/21 + 2\sin\theta\cos\theta = 2\Rightarrow \sin\theta\cos\theta = 1/2. → tan⁡θ+cot⁡θ=sin⁡2+cos⁡2sin⁡θcos⁡θ=1/(1/2)=2\tan\theta + \cot\theta = \dfrac{\sin^2 + \cos^2}{\sin\theta\cos\theta} = 1/(1/2) = 2.
Gyaan Maths · CBSE Class 10 · Chapter 8
Gyaan Maths — CBSE Class 6-10, built for board-toppers.
v1.1 · Feb 2026