Competitive Exams : Quantitative Aptitude - Trig (Set 1) | Concept of Maths

Question 1

1. Find the value of $\sin^2 30^\circ + \cos^2 30^\circ$.

A) 0

B) 1

C) $\frac{1}{2}$

D) 2

Answer

Correct Answer: B

Using the fundamental trigonometric identity: $\sin^2 \theta + \cos^2 \theta = 1$. Since $\theta = 30^\circ$ is the same for both, the value is $1$.

Question 2

2. If $\tan \theta = \frac{3}{4}$, then what is the value of $\sin \theta$?

A) $\frac{3}{5}$

B) $\frac{4}{5}$

C) $\frac{5}{3}$

D) 1

Answer

Correct Answer: A

We know $\tan \theta = \frac{\text{Perpendicular (P)}}{\text{Base (B)}} = \frac{3}{4}$. Using Pythagoras theorem: $H^2 = P^2 + B^2 \implies H = \sqrt{3^2 + 4^2} = 5$. Thus, $\sin \theta = \frac{P}{H} = \frac{3}{5}$.

Question 3

3. Evaluate the expression: $\frac{\sin 18^\circ}{\cos 72^\circ}$.

A) 0

B) 1

C) 2

D) $\frac{1}{2}$

Answer

Correct Answer: B

Using complementary angle formula: $\cos \theta = \sin(90^\circ - \theta)$. Therefore, $\cos 72^\circ = \sin(90^\circ - 72^\circ) = \sin 18^\circ$. The expression becomes $\frac{\sin 18^\circ}{\sin 18^\circ} = 1$.

Question 4

4. What is the value of $\tan 45^\circ + \cot 45^\circ$?

A) 0

B) 1

C) 2

D) $\sqrt{2}$

Question 5

5. If $\cos A = \frac{1}{2}$, find the value of $\sec A$.

A) 1

B) 2

C) $\frac{1}{2}$

D) $\sqrt{2}$

Answer

Correct Answer: B

$\sec A$ is the reciprocal of $\cos A$. Therefore, $\sec A = \frac{1}{\cos A} = \frac{1}{1/2} = 2$.

Question 6

6. The product $\sin \theta \cdot \csc \theta$ is always equal to:

A) 0

B) 1

C) $\sin^2 \theta$

D) $\cos \theta$

Answer

Correct Answer: B

By definition, $\csc \theta = \frac{1}{\sin \theta}$. Thus, $\sin \theta \cdot \frac{1}{\sin \theta} = 1$.

Question 7

7. Which of the following is equal to $(\sec^2 \theta - 1)$?

A) $\sin^2 \theta$

B) $\cos^2 \theta$

C) $\tan^2 \theta$

D) $\cot^2 \theta$

Answer

Correct Answer: C

From the identity $1 + \tan^2 \theta = \sec^2 \theta$, we can rearrange it as $\tan^2 \theta = \sec^2 \theta - 1$.

Question 8

8. A pole 6m high casts a shadow $2\sqrt{3}$m long on the ground. The angle of elevation of the sun is:

A) $30^\circ$

B) $45^\circ$

C) $60^\circ$

D) $90^\circ$

Answer

Correct Answer: C

$\tan \theta = \frac{\text{Height}}{\text{Shadow}} = \frac{6}{2\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}$. Since $\tan 60^\circ = \sqrt{3}$, the angle is $60^\circ$.

Question 9

9. Find the value of $\cos 0^\circ \cdot \cos 1^\circ \cdot \cos 2^\circ \dots \cos 90^\circ$.

A) 1

B) 0

C) -1

D) $\frac{1}{2}$

Answer

Correct Answer: B

The product contains the term $\cos 90^\circ$. Since $\cos 90^\circ = 0$, the entire product becomes 0.

Question 10

10. Evaluate: $\sin 60^\circ \cos 30^\circ + \cos 60^\circ \sin 30^\circ$.

A) $\frac{1}{2}$

B) 1

C) 0

D) $\frac{\sqrt{3}}{2}$

Answer

Correct Answer: B

This follows the formula $\sin(A+B) = \sin A \cos B + \cos A \sin B$. Here $A=60^\circ, B=30^\circ$. So, $\sin(60^\circ+30^\circ) = \sin 90^\circ = 1$.

Accepted Answer

Correct Answer: C

Competitive Exams : Quantitative Aptitude - Trig (Set 1) | Concept of Maths