NCERT solutions for class 10 maths chapter 8, Introduction to Trigonometry, live under this directory. The chapter builds three ideas: the six ratios of a right triangle, the exact values of the standard angles, and the three Pythagorean identities. Each idea has its own exercise.
This page tells you which exercise your homework belongs to — and which method it wants — before you open a solutions page.
There are three exercises: 8.1 (ratios), 8.2 (standard angles) and 8.3 (identities). Below the directory you’ll find the method map — the four techniques the whole chapter runs on — then the mistakes its questions are built to catch, and a short route for homework.
NCERT Solutions for Class 10 Maths Chapter 8: Exercise Directory
Every question of the chapter sits in one of three exercises. Pick by what your question gives you — a side pair, a single ratio, a standard-angle expression, or an identity to prove.
Exercise 8.1 — The Six Ratios and Recovering Every Ratio from One (11 questions)
This is the definitions and Pythagoras exercise. Question 1 asks for sin and cos of both angle A and angle C in the same triangle, so you see that the opposite and adjacent sides swap when the angle changes. Question 2 reads the side lengths off Fig. 8.13 to build \( \tan P – \cot R \).
Questions 3–5 and 10 hand you one ratio — \( \sin A = \frac{3}{4} \), \( \cot A \), \( \sec \theta \), or a side-sum — and demand every other ratio; these are the k-method questions. Questions 6–9 fold the ratios into comparisons and compound expressions, and Question 11 is a true/false check on which ratios can exceed 1.
Method: label opposite, hypotenuse and adjacent correctly, then apply Pythagoras.
Exercise 8.2 — Standard Angles and Table 8.1 (4 questions)
Question 1 substitutes the exact values of 0°, 30°, 45°, 60° and 90° into five compound expressions — sums, products and divisions of sin, cos, tan, sec and cosec. Question 2 offers four MCQs, each hiding a standard-angle value inside a form like \( \frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} \).
Question 3 turns \( \tan (A+B) = \sqrt{3} \) and \( \tan (A-B) = \frac{1}{\sqrt{3}} \) into two simultaneous equations and solves them for A and B. Question 4 is true/false on how sin and cos move as the angle grows, and on undefined cases.
Method: memorise Table 8.1, substitute the values, then solve.
Exercise 8.3 — Trigonometric Identities (4 questions, Q4 has 10 parts)
Questions 1–2 rewrite every other ratio in terms of a single given ratio — first \( \cot A \), then \( \sec A \) — using the identities. Question 3 applies \( \sin^2 A + \cos^2 A = 1 \), \( 1 + \tan^2 A = \sec^2 A \) and \( 1 + \cot^2 A = \cosec^2 A \) to four MCQs.
Question 4(i)–(x) is ten identity proofs. Each needs one of three moves: convert everything to sin and cos, apply the Pythagorean identity, or simplify the left hand side and right hand side separately. The textbook prints hints on (iii), (iv) and (ix).
Method: identity manipulation — pick the move the expression suggests.
Methods and Key Results the Whole Chapter Runs On
1. Side labelling
For any acute angle, the side that faces it is the opposite side, the side across the right angle is the hypotenuse, and the remaining side is the adjacent side (Fig. 8.4). Get these three wrong and every ratio that follows is wrong.
2. The k-method
The k-method recovers all six ratios when only one is known. Set the two sides named in the given ratio to \( k \) and a multiple of \( k \), apply Pythagoras to find the third side, then write each ratio by its definition. The chapter demonstrates it in Example 1 (p. 118).
3. Table 8.1 for standard angles
The values of sin, cos and tan at 0°, 30°, 45°, 60° and 90° must be memorised. The 30° and 60° rows come from halving an equilateral triangle (Fig. 8.15); the 45° row from splitting a square.
| \( \angle A \) | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin A | 0 | \( \frac{1}{2} \) | \( \frac{1}{\sqrt{2}} \) | \( \frac{\sqrt{3}}{2} \) | 1 |
| cos A | 1 | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{\sqrt{2}} \) | \( \frac{1}{2} \) | 0 |
| tan A | 0 | \( \frac{1}{\sqrt{3}} \) | 1 | \( \sqrt{3} \) | not defined |
Notice the trend: as the angle grows from 0° to 90°, sin rises from 0 to 1 while cos falls from 1 to 0. This monotonic behaviour is exactly what Exercise 8.2 Question 4 tests.
4. The three identities
Dividing the Pythagoras relation \( AB^2 + BC^2 = AC^2 \) by each side in turn gives the three identities every proof in this chapter leans on.
\[ \cos^2 A + \sin^2 A = 1 \quad \text{for } 0^\circ \leq A \leq 90^\circ \]
\[ 1 + \tan^2 A = \sec^2 A \quad \text{for } 0^\circ \leq A \lt 90^\circ \]
\[ 1 + \cot^2 A = \cosec^2 A \quad \text{for } 0^\circ \lt A \leq 90^\circ \]
Each identity carries a domain caveat: \( \tan A \) and \( \sec A \) divide by zero at 90°, so the second identity stops just short of it, while \( \cot A \) and \( \cosec A \) divide by zero at 0°.
One remark to file away: \( \sin A \) means ‘sine of angle A’, never sin × A, and \( \sin^{-1} A \) is not \( \cosec A \). These two confusions cost marks every year.
Common Mistakes in Chapter 8 and How to Fix Them
These are the errors the chapter’s questions are built to catch. Fix each one before you start an exercise.
- Treating sin A as sin × A. ‘sin’ separated from A has no meaning; it is a function applied to the angle.
- Confusing \( \sin^{-1} A \) with \( \cosec A \). They are different things: \( \cosec A = \frac{1}{\sin A} \), while \( \sin^{-1} A \) is a notation for inverse sine you meet in higher classes.
- Keeping the same opposite and adjacent sides when the angle changes from A to C. The sides swap — which is exactly what Exercise 8.1 Question 1 checks.
- Believing every ratio stays under 1. Only sin and cos are capped at 1, because the hypotenuse is the longest side. Tan, sec, cosec and cot can exceed 1 — the point of Exercise 8.1 Question 11.
- Writing tan 90°, sec 90°, cot 0° or cosec 0° as if they exist. Their definitions divide by zero, so Table 8.1 marks them ‘not defined’ (Exercise 8.2 Question 4 confirms).
- Leaving surds unrationalised. Convert \( \frac{1}{\sqrt{2}} \) to \( \frac{\sqrt{2}}{2} \) in a final answer.
How to Use This Page for Homework and Revision
Match your question to its exercise in four quick steps.
- Read your question and note what it gives you: a pair of sides, a single ratio, a standard-angle expression, or an identity to prove.
- Match it to Exercise 8.1 (ratios), Exercise 8.2 (standard angles) or Exercise 8.3 (identities) using the directory rows above.
- Open that exercise’s solutions page — each carries full step-by-step working.
- If you only need the key results, the chapter’s own summary (Section 8.5, p. 132) lists the ratios, the identities and the never-exceeds-1 rule.
Chapter 8 assumes the AA similarity criterion and Pythagoras from Chapter 6 (Triangles), and it is the direct foundation for Chapter 9, Some Applications of Trigonometry, where these ratios measure heights and distances. Chapter 7, Coordinate Geometry shares the same Pythagoras link. For the rest of the year’s maths, work from the Class 10 maths hub or the Class 10 hub.
When a value in Table 8.1 or one of the three identities looks doubtful, verify it against the official NCERT textbook PDF for Chapter 8 on ncert.nic.in — the ratios, the table and the identities are printed there exactly as the board tests them.
| Exercise | What you get |
|---|---|
| Exercise 8.1 | Solved |
| Exercise 8.2 | Solved |
| Exercise 8.3 (coming soon) | Coming soon |
Frequently Asked Questions on Chapter 8 Trigonometry
Why is tan 90° not defined in Class 10 trigonometry?
Because \( \tan A = \frac{\sin A}{\cos A} \) and \( \cos 90^\circ = 0 \), so the definition asks you to divide by zero. In the triangle itself, as angle A approaches 90° the adjacent side collapses towards zero, leaving the ratio without a value.
Will sin A or cos A ever be greater than 1 in this chapter?
No. Both are ratios with the hypotenuse — the longest side — in the denominator, so each stays between 0 and 1. Tan, sec, cosec and cot carry no such cap, and several of them exceed 1.
Which exercise in Chapter 8 has the identity proofs?
Exercise 8.3. Question 4 holds all ten proofs, labelled (i) to (x), each to be shown true for acute angles where the expressions are defined.
How many questions are there in Exercise 8.1 of Chapter 8?
Eleven. They run from finding two ratios at once, through the k-method questions, to the true/false list on which ratios can exceed 1.
What is the k method used for in trigonometric ratio problems?
When you are given one ratio, the k-method sets the two sides it names to \( k \) and a multiple of \( k \), uses Pythagoras to get the third side, then writes all six ratios by their definitions.
Reference: NCERT textbooks (CBSE).