Question 1

1. A ladder leans against a vertical wall at an angle of $60^\circ$ with the ground. If the foot of the ladder is 5 m away from the wall, what is the length of the ladder?

A) 5 mB) 10 mC) $5\sqrt{3}m$D) $10\sqrt{3}m$

Answer

✓ Correct Answer: Option B

Explanation: Solution: Let the length of the ladder be L. The distance from the wall (base) is 5 m. Using the cosine ratio: $$\cos 60^\circ = \frac{Base}{Hypotenuse}$ $\frac{1}{2} = \frac{5}{L}$ $L = 5 \times 2 = 10m$$.

Question 2

2. What is the value of the expression $\cos 1^\circ \cdot \cos 2^\circ \cdot \cos 3^\circ \cdot ... \cdot \cos 100^\circ$?

A) 1B) -1 $\frac{1}{\sqrt{2}}$

Question 3

3. If $\tan A = \cot B$, where A and B are acute angles, what is the value of (A + B)?

A) $90^\circ$B) $45^\circ$C) $180^\circ$D) $0^\circ$

Answer

✓ Correct Answer: Option A

Explanation: Solution: We know the complementary identity: $\cot B = \tan(90^\circ - B)$. Given $\tan A = \cot B$, we can write: $$\tan A = \tan(90^\circ - B)$ Comparing the angles: $A = 90^\circ - B ⇒ A + B = 90^\circ$$.

Question 4

4. Find the value of $\frac{\sin 18^\circ}{\cos 72^\circ} + \frac{\tan 25^\circ}{\cot 65^\circ}$.

A) 0B) 1C) 2D) 4

Answer

✓ Correct Answer: Option C

Explanation: Solution: Using complementary angles: $\cos 72^\circ = \cos(90^\circ - 18^\circ) = \sin 18^\circ$. $\cot 65^\circ = \cot(90^\circ - 25^\circ) = \tan 25^\circ$. Substituting these in the expression: $$\frac{\sin 18^\circ}{\sin 18^\circ} + \frac{\tan 25^\circ}{\tan 25^\circ} = 1 + 1 = 2$$.

Question 5

5. If $\sec \theta + \tan \theta = 3$, find the value of $\sec \theta - \tan \theta$.

A) 3B) 1/3C) -3D) 0

Answer

✓ Correct Answer: Option B

Explanation: Solution: We know the identity: $\sec^2 \theta - \tan^2 \theta = 1$. This can be factored as $(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1$. Given $(\sec \theta + \tan \theta) = 3$, we have: $$3 \times (\sec \theta - \tan \theta) = 1$ $\sec \theta - \tan \theta = \frac{1}{3}$$.

Question 6

6. The shadow of a vertical pole is equal to its height. What is the angle of elevation of the sun?

A) $30^\circ$B) $60^\circ$C) $90^\circ$D) $45^\circ$

Answer

✓ Correct Answer: Option D

Explanation: Solution: Let the height of the pole be h and the length of the shadow be s. Given h = s. If $\theta$ is the angle of elevation: $$\tan \theta = \frac{Height}{Shadow} = \frac{h}{s}$ Sinceh = s,\tan \theta = 1. We know\tan 45^\circ = 1, so\theta = 45^\circ$.

Question 7

7. Simplify the expression: $(\sin^2 \theta + \cos^2 \theta) \cdot (\sec^2 \theta - \tan^2 \theta)$.

A) 1B) 0C) 2D) -1

Answer

✓ Correct Answer: Option A

Explanation: Solution: We use basic trigonometric identities: 1. $\sin^2 \theta + \cos^2 \theta = 1$ 2. $\sec^2 \theta - \tan^2 \theta = 1$ Therefore: $$1 \times 1 = 1$$.

Question 8

8. Convert $60^\circ$ into radians.

A) $\frac{\pi}{2}$B) $\frac{\pi}{4}$C) $\frac{\pi}{3}$D) $\frac{\pi}{6}$

Answer

✓ Correct Answer: Option C

Explanation: Solution: To convert degrees to radians, multiply by $\frac{\pi}{180}$. $$Radians = 60 \times \frac{\pi}{180}$ $Radians = \frac{\pi}{3}$$.

Question 9

9. If $x = 5 \sin \theta$ and $y = 5 \cos \theta$, what is the value of $x^2 + y^2$?

A) 5B) 25C) 1D) 10

Answer

✓ Correct Answer: Option B

Explanation: Solution: Square both equations and add them: $x^2 = 25 \sin^2 \theta$ $y^2 = 25 \cos^2 \theta$ $$x^2 + y^2 = 25 \sin^2 \theta + 25 \cos^2 \theta$ $x^2 + y^2 = 25 (\sin^2 \theta + \cos^2 \theta)$ Since\sin^2 \theta + \cos^2 \theta = 1: $x^2 + y^2 = 25(1) = 25$$.

Question 10

10. Find the maximum value of $2 + \sin \theta$.

A) 3B) 2C) 1D) 0

Answer

✓ Correct Answer: Option A

Explanation: Solution: The range of $\sin \theta$ is [-1, 1]. The maximum value of $\sin \theta$ is 1. Therefore, the maximum value of $2 + \sin \theta$ is: $2 + 1 = 3$.

Accepted Answer

✓ Correct Answer: Option D

Explanation: Solution: The series includes the term $\cos 90^\circ$. We know that $\cos 90^\circ = 0$. Since all terms are multiplied together, the presence of a zero makes the entire product zero. $$Product = (\cos 1^\circ \cdot ... \cdot \cos 89^\circ) \times 0 \times (\cos 91^\circ \cdot ... \cdot \cos 100^\circ) = 0$$.

Competitive Exams : Quantitative Aptitude - Trig (Set 3)

1. A ladder leans against a vertical wall at an angle of $60^\circ$ with the ground. If the foot of the ladder is 5 m away from the wall, what is the length of the ladder?

2. What is the value of the expression $\cos 1^\circ \cdot \cos 2^\circ \cdot \cos 3^\circ \cdot ... \cdot \cos 100^\circ$?

3. If $\tan A = \cot B$, where A and B are acute angles, what is the value of (A + B)?

4. Find the value of $\frac{\sin 18^\circ}{\cos 72^\circ} + \frac{\tan 25^\circ}{\cot 65^\circ}$.

5. If $\sec \theta + \tan \theta = 3$, find the value of $\sec \theta - \tan \theta$.

6. The shadow of a vertical pole is equal to its height. What is the angle of elevation of the sun?

7. Simplify the expression: $(\sin^2 \theta + \cos^2 \theta) \cdot (\sec^2 \theta - \tan^2 \theta)$.

8. Convert $60^\circ$ into radians.

9. If $x = 5 \sin \theta$ and $y = 5 \cos \theta$, what is the value of $x^2 + y^2$?

10. Find the maximum value of $2 + \sin \theta$.