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CBSE - Class 10 Mathematics Introduction to Trigonometry Worksheet

If $\tan (A + B) = \sqrt{3}$ and $\tan (A – B) = \frac{1}{\sqrt{3}}$; $0^\circ < A + B \le 90^\circ$; $A > B$, find $A$ and $B$.

Choose the correct option and justify your choice : (iv) $\frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} =$

$\cos 60^\circ$

$\sin 60^\circ$

$\tan 60^\circ$

$\sin 30^\circ$

In Fig. 8.13, find $\tan P – \cot R$.

Chapter 8 – Introduction to Trigonometry Questions and Answers ...

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ix) $(\text{cosec } A – \sin A) (\sec A – \cos A) = \frac{1}{\tan A + \cot A}$
[Hint : Simplify LHS and RHS separately]

State whether the following are true or false. Justify your answer. (iv) $\cot A$ is the product of cot and $A$.

State whether the following are true or false. Justify your answer. (v) $\cot A$ is not defined for $A = 0^\circ$.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) $(\text{cosec } \theta – \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ii) $\frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2 \sec A$

If $\sin A = \frac{3}{4}$, calculate $\cos A$ and $\tan A$.

10.

Choose the correct option and justify your choice : (i) $\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =$

$\sin 60^\circ$

$\cos 60^\circ$

$\tan 60^\circ$

$\sin 30^\circ$

11.

Evaluate the following : (v) $\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$

12.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iii) $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \text{cosec } \theta$
[Hint : Write the expression in terms of $\sin \theta$ and $\cos \theta$]

13.

Choose the correct option. Justify your choice. (iv) $\frac{1 + \tan^2 A}{1 + \cot^2 A} =$

$\sec^2 A$

–1

$\cot^2 A$

$\tan^2 A$

14.

In $\triangle ABC$, right-angled at $B$, $AB = 24$ cm, $BC = 7$ cm. Determine : (ii) $\sin C, \cos C$

15.

Evaluate the following : (iv) $\frac{\sin 30^\circ + \tan 45^\circ – \text{cosec } 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$

16.

If tanθ = 1, then value of (1 – sinθ)(1 + sinθ) = ?

1/2

17.

Evaluate the following : (iii) $\frac{\cos 45^\circ}{\sec 30^\circ + \text{cosec } 30^\circ}$

18.

If $\angle A$ and $\angle B$ are acute angles such that $\cos A = \cos B$, then show that $\angle A = \angle B$.

19.

Choose the correct option. Justify your choice. (i) $9 \sec^2 A – 9 \tan^2 A =$

20.

Given $\sec \theta = \frac{13}{12}$, calculate all other trigonometric ratios.

Worksheet Answers

10.

13.

16.

Option B

Solution:

Hinglish:
Step 1: tanθ = 1 → θ = 45°
Step 2: sin45° = √2/2
Step 3: (1 – sinθ)(1 + sinθ) = 1 – sin²θ = 1 – (√2/2)² = 1 – 2/4 = 1 – 1/2 = 1/2

Concept: Use algebraic identity a² – b² = (a – b)(a + b) aur basic trig value step by step apply karo.

English:
Step 1: tanθ = 1 → θ = 45°
Step 2: sin45° = √2/2
Step 3: (1 – sinθ)(1 + sinθ) = 1 – sin²θ = 1 – (√2/2)² = 1 – 1/2 = 1/2

Concept: Use a² – b² = (a – b)(a + b) and basic trig values step by step.

19.

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