CBSE-Class-10-Maths-Assertion-and-Reason-Chapter-9-Some-Applications-of-Trigonometry

Class 10^th Maths Chapter Assertion and Reason Questions

Question 1

Assertion (A): The height of an object can be calculated using trigonometry if the angle of elevation and distance from the observer are known.
Reason (R): The tangent of the angle of elevation is the ratio of the height of the object to the distance from the observer.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Using $\tan \theta = \frac{\text{height}}{\text{distance}}$ , one can calculate the height if the distance and angle are known.

Question 2

Assertion (A): If the angle of elevation of the top of a building from a point 50 m away is $30^\circ$ , then the height of the building is $25\sqrt{3}$ m.
Reason (R): The formula $\tan \theta = \frac{\text{height}}{\text{distance}}$ can be used to find the height.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: $\tan 30^\circ = \frac{\text{height}}{50} \Rightarrow \frac{1}{\sqrt{3}} = \frac{\text{height}}{50} \Rightarrow \text{height} = 50 \times \frac{1}{\sqrt{3}} = 25\sqrt{3}$ .

Question 3

Assertion (A): The angle of depression from a point on a cliff to a point on the ground is always less than $90^\circ$ .
Reason (R): Angles of elevation and depression are measured from the horizontal line.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The angle of depression is measured from the horizontal, so it must be less than $90^\circ$ .

Question 4

Assertion (A): In a right triangle, if one angle is $45^\circ$ , then the two legs of the triangle are equal in length.
Reason (R): The tangent of a $45^\circ$ angle is 1.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: $\tan 45^\circ = 1$ , so opposite side = adjacent side, meaning both legs are equal in length.

Question 5

Assertion (A): In any right triangle, the ratio of the opposite side to the hypotenuse is always less than 1.
Reason (R): Sine of any angle is a ratio of the opposite side to the hypotenuse, which has a maximum value of 1.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The hypotenuse is the longest side, so the ratio of the opposite side to the hypotenuse is always less than 1.

Question 6

Assertion (A): If the angle of elevation of an object increases, the distance between the observer and the object decreases.
Reason (R): An increase in the angle of elevation implies a decrease in the horizontal distance for the same height.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: As the angle of elevation increases, the observer needs to be closer to the object to maintain the same angle.

Question 7

Assertion (A): The angle of depression from a cliff is always equal to the angle of elevation from the ground level to the top of the cliff.
Reason (R): Angles of elevation and depression are congruent when they are measured from the same horizontal line.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: By alternate interior angles formed by a transversal line intersecting two parallel lines, the angles of elevation and depression are equal.

Question 8

Assertion (A): In a right triangle with one angle as $60^\circ$ , the opposite side to this angle is equal to $\frac{\sqrt{3}}{2}$ of the hypotenuse.
Reason (R): $\sin 60^\circ = \frac{\sqrt{3}}{2}$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: $\sin 60^\circ = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}$ , confirming the assertion.

Question 9

Assertion (A): For an angle $\theta$ in a right triangle, if $\tan \theta$ increases, then $\theta$ also increases.
Reason (R): The tangent function is an increasing function in the interval $0^\circ < \theta < 90^\circ$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Since $\tan\theta$ increases as $\theta$ increases in the interval $0^\circ < \theta < 90^\circ$ , $\tan\theta$ and $\theta$ are directly related.

Question 10

Assertion (A): In a right-angled triangle, the value of $\cos 90^\circ$ is zero.
Reason (R): The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: The cosine of $90^\circ$ is zero due to trigonometric properties, not because of the adjacent side to hypotenuse ratio.

Question 11

Assertion (A): The value of $\sin 30^\circ$ is $\frac{1}{2}$ .
Reason (R): The opposite side is half the hypotenuse when the angle is $30^\circ$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: In a $30^\circ$ angle, the side opposite is half the length of the hypotenuse.

Question 12

Assertion (A): $\tan 45^\circ = 1$ .
Reason (R): The ratio of opposite to adjacent sides for a $45^\circ$ angle in a right triangle is 1.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: For $45^\circ$ , opposite = adjacent, so $\tan 45^\circ = 1$ .

Question 13

Assertion (A): $\sin 90^\circ = 1$ .
Reason (R): The hypotenuse and the opposite side are equal for a $90^\circ$ angle.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: C) A is true, but R is false.
Explanation: $\sin 90^\circ$ is 1, but the hypotenuse cannot be equal to the opposite side in a right triangle.

Question 14

Assertion (A): In a right triangle, if one of the angles is $60^\circ$ , the other non-right angle is $30^\circ$ .
Reason (R): The sum of the angles in a triangle is $180^\circ$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: In a triangle, $90^\circ + 60^\circ = 150^\circ$ , so the remaining angle is $30^\circ$ .

Question 15

Assertion (A): The angle of elevation of the sun decreases as it moves towards the horizon.
Reason (R): The angle of elevation is the angle between the horizontal and a line to an object above.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: As the sun moves towards the horizon, its elevation angle decreases.

Question 16

Assertion (A): The cosine of an acute angle in a right triangle is always less than 1.
Reason (R): The adjacent side is always shorter than the hypotenuse in a right triangle.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Since the hypotenuse is the longest side, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} < 1$ .

Question 17

Assertion (A): For an angle $\theta$ , $\sec \theta$ is the reciprocal of $\cos \theta$ .
Reason (R): Trigonometric functions have reciprocal identities that allow calculation of other functions.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: $\sec \theta = \frac{1}{\cos \theta}$ is a reciprocal identity.

Question 18

Assertion (A): In a right triangle, the value of $\tan \theta$ is the ratio of the opposite side to the adjacent side.
Reason (R): The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in trigonometry.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: By definition, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ .

Question 19

Assertion (A): The sine of an angle is always positive in a right triangle.
Reason (R): In trigonometry, sine values for angles from $0^\circ$ to $90^\circ$ are positive.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: For acute angles, $\sin \theta$ is positive.

Question 20

Assertion (A): The value of $\cot 45^\circ$ is 1.
Reason (R): The cotangent function is the reciprocal of the tangent function.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: $\cot 45^\circ = \frac{1}{\tan 45^\circ} = 1$ .

Question 21

Assertion (A): If $\sin \theta = \frac{3}{5}$ , then $\cos \theta = \frac{4}{5}$ for the same right triangle.
Reason (R): Pythagorean identity can be used to find $\cos \theta$ when $\sin \theta$ is known.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Using Pythagorean theorem: $\sin^2 \theta + \cos^2 \theta = 1$ .

Question 22

Assertion (A): In a right triangle, $\sec 45^\circ = \sqrt{2}$ .
Reason (R): $\sec \theta$ is the reciprocal of $\cos \theta$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Since $\cos 45^\circ = \frac{1}{\sqrt{2}}$ , $\sec 45^\circ = \sqrt{2}$ .

Question 23

Assertion (A): The angles of elevation and depression are equal for an object at the same distance from an observer’s height.
Reason (R): The angle of elevation equals the angle of depression if the horizontal is the same.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Both angles are formed from parallel lines and are equal by alternate interior angles.

Question 24

Assertion (A): The height of an object can be found if the angle of elevation and distance from the object is known.
Reason (R): The trigonometric ratios involve the angle and sides of a right triangle.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Height can be determined by $\tan$ using distance and angle.

Question 25

Assertion (A): The distance of an object can be determined if the angle of depression and height of the object are known.
Reason (R): In right triangles, trigonometric functions relate angles and sides.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Using $\tan$ , distance can be found from the angle and known height.

Chapter 9 Some Applications of Trigonometry

Class 10th Maths Chapter Assertion and Reason Questions