CBSE Class 10 Maths — Introduction to Trigonometry

Chapter Overview & Weightage

Introduction to Trigonometry in Class 10 builds the foundation for all of Class 11-12 trigonometry and is directly useful in JEE preparation. The chapter covers trigonometric ratios, standard angles, complementary angles, and fundamental identities.

Trigonometry carries 8–10 marks in CBSE Class 10 board exams — one of the highest-weighted chapters. A typical distribution: 1 MCQ + 1 short answer (evaluate expression with standard angles) + 1 medium question (prove identity) + 1 long answer (application or complementary angles). Identity proofs and standard angle evaluations are must-practise topics.

Year	Marks	Key Questions
2024	10	MCQ, standard angle evaluation, prove identity, heights and distances
2023	9	Complementary angles, identity proof (3 marks), standard angle application
2022	8	Evaluate expression, prove using identities, word problem
2021	8	MCQ, prove identity, complementary angle result

Key Concepts You Must Know

1. Trigonometric Ratios (for a right triangle with angle θ):

\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \quad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}

\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}

2. Reciprocal Relations: $\sin\theta \cdot \csc\theta = 1$ ; $\cos\theta \cdot \sec\theta = 1$ ; $\tan\theta \cdot \cot\theta = 1$

3. Quotient Relations: $\tan\theta = \frac{\sin\theta}{\cos\theta}$ ; $\cot\theta = \frac{\cos\theta}{\sin\theta}$

4. Complementary Angles ( $\theta$ and $90° - \theta$ are complementary):

\sin(90° - \theta) = \cos\theta, \quad \cos(90° - \theta) = \sin\theta

\tan(90° - \theta) = \cot\theta, \quad \sec(90° - \theta) = \csc\theta

5. Trigonometric Identities (Pythagorean):

\sin^2\theta + \cos^2\theta = 1

1 + \tan^2\theta = \sec^2\theta

1 + \cot^2\theta = \csc^2\theta

Important Formulas

Angle	sin	cos	tan	cot	sec	csc
0°	0	1	0	undef	1	undef
30°	1/2	√3/2	1/√3	√3	2/√3	2
45°	1/√2	1/√2	1	1	√2	√2
60°	√3/2	1/2	√3	1/√3	2	2/√3
90°	1	0	undef	0	undef	1

\sin^2\theta + \cos^2\theta = 1

\sec^2\theta - \tan^2\theta = 1

\csc^2\theta - \cot^2\theta = 1

These are the three most frequently used identities in Class 10.

Solved Previous Year Questions

PYQ 1 — Evaluate (CBSE 2023, 2 marks)

\frac{\cos 37°}{\sin 53°} + \frac{\sin 41°}{\cos 49°}

Solution:

Note: $53° = 90° - 37°$ , so $\sin 53° = \cos 37°$ . Note: $49° = 90° - 41°$ , so $\cos 49° = \sin 41°$ .

= \frac{\cos 37°}{\cos 37°} + \frac{\sin 41°}{\sin 41°} = 1 + 1 = \mathbf{2}

PYQ 2 — Prove Identity (CBSE 2024, 3 marks)

Prove: $\frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta} = 2\csc\theta$

Solution:

LHS = $\frac{\sin^2\theta + (1+\cos\theta)^2}{\sin\theta(1+\cos\theta)}$

Numerator: $\sin^2\theta + 1 + 2\cos\theta + \cos^2\theta = (\sin^2\theta + \cos^2\theta) + 1 + 2\cos\theta = 1 + 1 + 2\cos\theta = 2 + 2\cos\theta = 2(1+\cos\theta)$

LHS = $\frac{2(1+\cos\theta)}{\sin\theta(1+\cos\theta)} = \frac{2}{\sin\theta} = 2\csc\theta$ = RHS ✓

PYQ 3 — Standard Angles (2 marks)

Evaluate: $\sin^2 60° + 2\tan 45° - \cos^2 30°$

= \left(\frac{\sqrt{3}}{2}\right)^2 + 2(1) - \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} + 2 - \frac{3}{4} = 2

Difficulty Distribution

Difficulty	Topics	Approximate %
Easy	Standard angle evaluation, reciprocal/quotient relations	30%
Medium	Complementary angles, simple identity proofs	45%
Hard	Multi-step identity proofs, finding all ratios given one ratio	25%

Expert Strategy

Memorise the standard angle table completely. It directly answers 30–40% of trigonometry problems. The “sin increases, cos decreases from 0° to 90°” pattern helps reconstruct it: sin values are $0, \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, 1$ for $0°, 30°, 45°, 60°, 90°$ .

For identity proofs: Always work on one side only (usually the more complex side). The goal is to transform it into the other side using known identities.

Complementary angle shortcut: When you see angles like $37°$ and $53°$ , or $\sin 20°/\cos 70°$ , immediately apply the complementary angle relation. The sum is always $90°$ .

The mnemonic “SOHCAHTOA” (Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent) helps recall the basic definitions. Also: “Some People Have Curly Brown Hair Till Painted Black” → Sin, P (perpendicular), H (hypotenuse) / Cos, B (base), H / Tan, P, B.

Common Traps

Trap 1: Writing $\sin 2\theta = 2\sin\theta$ . This is WRONG. $\sin 2\theta = 2\sin\theta\cos\theta$ (double angle formula). For Class 10, double angle formulas aren’t needed, but don’t make arithmetic errors like $\sin 60° = 2\sin 30°$ — verify: $\frac{\sqrt{3}}{2} \neq 2 \times \frac{1}{2} = 1$ .

Trap 2: Assuming $\sin\theta + \cos\theta = 1$ . The correct identity is $\sin^2\theta + \cos^2\theta = 1$ . The sum $\sin\theta + \cos\theta$ can range from $-\sqrt{2}$ to $\sqrt{2}$ . This is the most common identity confusion in Class 10.

Trap 3: Dividing by zero in identity proofs. When simplifying, students sometimes cancel terms that could be zero. For example, dividing both numerator and denominator by $\cos\theta$ without noting that $\cos\theta \neq 0$ for the identity to hold. In board exams, briefly note the restriction (e.g., “valid for $\theta \neq 90°$ ”).

Trap 4: tan 90° and cot 0° are undefined. Any problem involving these values has no answer, or the question is designed to avoid them. If your simplification leads to $\tan 90°$ , recheck your steps.