Textbook chapter 8
Relate an acute angle to side ratios in a right triangle, evaluate standard angles and simplify expressions using identities.
Textbook exercise solutions
- Exercise 8.1 — 11 question groups
- Exercise 8.2 — 4 question groups
- Exercise 8.3 — 4 question groups
Videos, notes and tests
English · Session 2026–27 · NCERT-aligned study guide
Chapter overview
Relate an acute angle to side ratios in a right triangle, evaluate standard angles and simplify expressions using identities.
Before you begin: Right triangles, similar triangles, Pythagoras and fractions.
Key concepts · Solved examples · Practice · Mind map · Timed chapter test
Learn the key concepts
Choosing the sides
Relative to a chosen acute angle θ, mark the opposite side O, adjacent leg A and hypotenuse H. The hypotenuse is opposite the right angle. Similar right triangles with the same θ have proportional sides, so the ratios depend on θ rather than triangle size.
Ratios and special angles
sinθ=O/H, cosθ=A/H, tanθ=O/A. Their reciprocals are cosecθ, secθ and cotθ where denominators are nonzero. At 30°,45°,60°, sine values are 1/2,1/√2,√3/2; cosine values are √3/2,1/√2,1/2; tangent values are 1/√3,1,√3. At 0°, sin=0 and cos=1; at 90°, sin=1 and cos=0. tan90°,sec90°,cosec0° and cot0° are undefined.
Proving identities
Divide O²+A²=H² by H² to obtain sin²θ+cos²θ=1. Dividing that identity by cos²θ or sin²θ gives two more identities where defined. For a proof, transform one side logically into the other; do not assume the statement you are trying to prove.
Formula and result checklist
- sin²θ+cos²θ=1.
- 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ.
- tanθ=sinθ/cosθ where cosθ≠0.
Solved examples
Example 1
If sinθ=5/13 and θ is acute, find cosθ and tanθ.
Choose O=5,H=13. A=√(169−25)=12. Thus cosθ=12/13 and tanθ=5/12.
Example 2
Evaluate 2sin30°+cos60°−tan45°.
Substitute 2(1/2)+(1/2)−1 = 1/2.
Example 3
Prove (1−cos²θ)/sinθ=sinθ where sinθ≠0.
Replace 1−cos²θ by sin²θ. Then sin²θ/sinθ=sinθ, as required. The nonzero condition allows cancellation.
Try it yourself
Work on paper before opening each answer. Show the reasoning and include units when needed.
1. If tanθ=3/4 for acute θ, find sinθ.
Check answer 1
Opposite=3, adjacent=4, hypotenuse=5; sinθ=3/5.
2. Evaluate sin²60°+cos²60°.
Check answer 2
1.
3. Evaluate sec²45°−tan²45°.
Check answer 3
1; 2−1=1.
Common mistakes to avoid
- sin²θ means (sinθ)², not sin(θ²).
- Do not assign a finite value to tan90°.
Chapter mind map
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Take the 6-question, 10-minute chapter practice test. Each correct answer earns 1 mark; there is no negative marking. See explanations after submission.
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Read the corresponding chapter in the official NCERT textbook, then complete its exercises on paper. These NSL explanations and original practice questions support your textbook study; they are not an exercise-by-exercise solution manual.
Official CBSE syllabus 2026–27 · All chapters · Previous chapter · Next chapter
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