CBSE • Class 10 • Mathematics • Chapter 8 (Introduction to Trigonometry) • Exercise 8.2

Exercise 8.2: Introduction to Trigonometry — NCERT Solutions

Trigonometric ratios at standard angles 0°, 30°, 45°, 60°, 90°.

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What this exercise covers

Standard-angle tableEvaluating expressionsSimplification

Step-by-step solutions — Exercise 8.2

3 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 8.2 Q1 • 2 marks

Evaluate: sin⁡60∘cos⁡30∘+sin⁡30∘cos⁡60∘\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ.
Hint (Socratic — try this first)▾
Do you recall the exact standard values of sine and cosine at 30∘30^\circ and 60∘60^\circ?
Step-by-step solution▾

Using standard values: sin⁡60∘=32,cos⁡30∘=32,sin⁡30∘=12,cos⁡60∘=12\sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}, \quad \cos 60^\circ = \frac{1}{2}

Substitute: sin⁡60∘cos⁡30∘+sin⁡30∘cos⁡60∘=32⋅32+12⋅12\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ = \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{1}{2}\cdot\frac{1}{2} =34+14=44=1= \frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1

(This is in fact sin⁡(60∘+30∘)=sin⁡90∘=1\sin(60^\circ + 30^\circ) = \sin 90^\circ = 1.)

Common mistake:
Mixing up the values, e.g. writing sin⁡60∘=12\sin 60^\circ = \frac{1}{2} instead of 32\frac{\sqrt3}{2}.
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Exercise 8.2 Q2 • 3 marks

Evaluate 2tan⁡245∘+cos⁡230∘−sin⁡260∘cot⁡245∘\dfrac{2\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ}{\cot^2 45^\circ}.
Hint (Socratic — try this first)▾
What are tan⁡45∘\tan 45^\circ and cot⁡45∘\cot 45^\circ, and do cos⁡30∘\cos 30^\circ and sin⁡60∘\sin 60^\circ have equal values here?
Step-by-step solution▾

Standard values: tan⁡45∘=1,cot⁡45∘=1,cos⁡30∘=32,sin⁡60∘=32\tan 45^\circ = 1, \quad \cot 45^\circ = 1, \quad \cos 30^\circ = \frac{\sqrt3}{2}, \quad \sin 60^\circ = \frac{\sqrt3}{2}

Numerator: 2tan⁡245∘+cos⁡230∘−sin⁡260∘=2(1)2+(32)2−(32)22\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ = 2(1)^2 + \left(\frac{\sqrt3}{2}\right)^2 - \left(\frac{\sqrt3}{2}\right)^2 =2+34−34=2= 2 + \frac{3}{4} - \frac{3}{4} = 2

Denominator: cot⁡245∘=12=1\cot^2 45^\circ = 1^2 = 1

Therefore the value =21=2= \dfrac{2}{1} = 2.

Common mistake:
Squaring incorrectly, e.g. taking cos⁡230∘=32\cos^2 30^\circ = \frac{\sqrt3}{2} instead of 34\frac{3}{4}.
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Exercise 8.2 Q3 • 3 marks

If tan⁡(A+B)=3\tan(A + B) = \sqrt{3} and tan⁡(A−B)=13\tan(A - B) = \dfrac{1}{\sqrt3}, where 0∘<A+B≤90∘0^\circ < A + B \le 90^\circ and A>BA > B, find AA and BB.
Hint (Socratic — try this first)▾
Which standard angles give tan⁡\tan equal to 3\sqrt3 and 13\frac{1}{\sqrt3}?
Step-by-step solution▾

We know: tan⁡60∘=3  ⟹  A+B=60∘(1)\tan 60^\circ = \sqrt3 \implies A + B = 60^\circ \quad (1) tan⁡30∘=13  ⟹  A−B=30∘(2)\tan 30^\circ = \frac{1}{\sqrt3} \implies A - B = 30^\circ \quad (2)

Adding (1) and (2): 2A=90∘  ⟹  A=45∘2A = 90^\circ \implies A = 45^\circ

Substituting into (1): 45∘+B=60∘  ⟹  B=15∘45^\circ + B = 60^\circ \implies B = 15^\circ

Hence A=45∘A = 45^\circ and B=15∘B = 15^\circ.

Common mistake:
Matching tan⁡(A+B)=3\tan(A+B)=\sqrt3 to the wrong angle (e.g. 30∘30^\circ) because of confusing tan⁡30∘\tan 30^\circ and tan⁡60∘\tan 60^\circ.
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How to approach Exercise 8.2

  1. Re-read the chapter summary first. Open Introduction to Trigonometry and refresh the key concepts: Trigonometric ratios, Standard angles, Pythagorean identity, Reciprocal identities.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Introduction to Trigonometry

  1. Exercise 8.1 — Trigonometric ratios for an acute angle in a right triangle.
  2. Exercise 8.2 — Trigonometric ratios at standard angles 0°, 30°, 45°, 60°, 90°.
  3. Exercise 8.3 — Trigonometric identities — sin²θ + cos²θ = 1 and consequences.

← Exercise 8.1

Trigonometric ratios for an acute angle in a right triangle.

Exercise 8.3 →

Trigonometric identities — sin²θ + cos²θ = 1 and consequences.

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