ROUTERA

Chapter 8 Introduction to Trigonometry

Class 10th Maths Chapter Assertion and Reason Questions

Question 1

Assertion (A): The value of sin⁡45∘\sin 45^\circ is 22\frac{\sqrt{2}}{2}.
Reason (R): The value of sin⁡45∘\sin 45^\circ is calculated as 12\frac{1}{\sqrt{2}}, which simplifies to 22\frac{\sqrt{2}}{2} when rationalized.

  • 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 value of sin⁡45∘\sin 45^\circ is indeed 22\frac{\sqrt{2}}{2}, and this is obtained by rationalizing 12\frac{1}{\sqrt{2}}.

Question 2

Assertion (A): The value of cos⁡30∘\cos 30^\circ is 32\frac{\sqrt{3}}{2}.
Reason (R): The cosine of an angle in a right triangle represents 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 value of cos⁡30∘\cos 30^\circ is 32\frac{\sqrt{3}}{2}. Although the reason correctly describes cos⁡θ\cos\theta, it doesn’t explain the specific value.

Question 3

Assertion (A): sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 for all values of θ\theta.
Reason (R): This identity is derived from the Pythagorean theorem applied to 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: The identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 is indeed derived from the Pythagorean theorem in a right triangle.

Question 4

Assertion (A): The value of tan⁡45∘\tan 45^\circ is 1.
Reason (R): The tangent of an angle is the ratio of the opposite side to the adjacent side 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: B) Both A and R are true, but R is not the correct explanation of A.
  • Explanation: The value of tan⁡45∘=1\tan 45^\circ = 1, but this is because in a 45° angle triangle, opposite and adjacent sides are equal, not directly because of the definition of tangent.

Question 5

Assertion (A): The value of sec⁡60∘\sec 60^\circ is 2.
Reason (R): The value of cos⁡60∘\cos 60^\circ is 12\frac{1}{2}, and sec⁡θ\sec \theta is the reciprocal of cos⁡θ\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⁡60∘=12\cos 60^\circ = \frac{1}{2}, sec⁡60∘=1cos⁡60∘=2\sec 60^\circ = \frac{1}{\cos 60^\circ} = 2.

Question 6

Assertion (A): cot⁡90∘\cot 90^\circ is undefined.
Reason (R): Cotangent is the reciprocal of tangent, and tan⁡90∘\tan 90^\circ is undefined.

  • 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⁡90∘\tan 90^\circ is undefined, so its reciprocal, cot⁡90∘\cot 90^\circ, is also undefined.

Question 7

Assertion (A): sin⁡0∘=0\sin 0^\circ = 0.
Reason (R): The sine function represents the y-coordinate of a point on the unit circle at the given 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: A) Both A and R are true, and R is the correct explanation of A.
  • Explanation: At 0∘0^\circ, the point on the unit circle has a y-coordinate of 0, thus sin⁡0∘=0\sin 0^\circ = 0.

Question 8

Assertion (A): The value of tan⁡60∘\tan 60^\circ is 3\sqrt{3}.
Reason (R): tan⁡θ\tan \theta represents the ratio of sin⁡θ\sin \theta to cos⁡θ\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: B) Both A and R are true, but R is not the correct explanation of A.
  • Explanation: tan⁡60∘=sin⁡60∘cos⁡60∘=3/21/2=3\tan 60^\circ = \frac{\sin 60^\circ}{\cos 60^\circ} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}. Though the reason is true, it doesn't explain the specific value.

Question 9

Assertion (A): For any angle θ\theta, sin⁡(90∘−θ)=cos⁡θ\sin(90^\circ - \theta) = \cos \theta.
Reason (R): Sine and cosine are co-functions of each other.

  • 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: This identity shows that sine and cosine values are related as co-functions.

Question 10

Assertion (A): sec⁡0∘=1\sec 0^\circ = 1.
Reason (R): The value of cos⁡0∘=1\cos 0^\circ = 1, and sec⁡θ\sec \theta is the reciprocal of cos⁡θ\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 sec⁡0∘=1cos⁡0∘=1\sec 0^\circ = \frac{1}{\cos 0^\circ} = 1.

Question 11

Assertion (A): The value of cot⁡45∘\cot 45^\circ is 1.
Reason (R): Cotangent is the reciprocal of tangent, and tan⁡45∘=1\tan 45^\circ = 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: Since cot⁡θ\cot \theta is the reciprocal of tan⁡θ\tan \theta, cot⁡45∘=1tan⁡45∘=1\cot 45^\circ = \frac{1}{\tan 45^\circ} = 1.

Question 12

Assertion (A): sin⁡30∘+cos⁡60∘=1\sin 30^\circ + \cos 60^\circ = 1.
Reason (R): sin⁡30∘=cos⁡60∘=12\sin 30^\circ = \cos 60^\circ = \frac{1}{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⁡30∘=12\sin 30^\circ = \frac{1}{2} and cos⁡60∘=12\cos 60^\circ = \frac{1}{2}, so their sum is 11.

Question 13

Assertion (A): sec⁡30∘=23\sec 30^\circ = \frac{2}{\sqrt{3}}.
Reason (R): sec⁡θ\sec \theta is the reciprocal of cos⁡θ\cos \theta, and cos⁡30∘=32\cos 30^\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: Since sec⁡30∘=1cos⁡30∘\sec 30^\circ = \frac{1}{\cos 30^\circ}, and cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}, sec⁡30∘=23\sec 30^\circ = \frac{2}{\sqrt{3}}.

Question 14

Assertion (A): The value of tan⁡0∘\tan 0^\circ is 0.
Reason (R): Tangent is the ratio of sine to cosine, and sin⁡0∘=0\sin 0^\circ = 0.

  • 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⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}, tan⁡0∘=sin⁡0∘cos⁡0∘=0\tan 0^\circ = \frac{\sin 0^\circ}{\cos 0^\circ} = 0.

Question 15

Assertion (A): csc⁡90∘=1\csc 90^\circ = 1.
Reason (R): Cosecant is the reciprocal of sine, and sin⁡90∘=1\sin 90^\circ = 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: csc⁡90∘=1sin⁡90∘=1\csc 90^\circ = \frac{1}{\sin 90^\circ} = 1.

Question 16

Assertion (A): sin⁡60∘+sin⁡30∘=3+12\sin 60^\circ + \sin 30^\circ = \frac{\sqrt{3} + 1}{2}.
Reason (R): sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2} and sin⁡30∘=12\sin 30^\circ = \frac{1}{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: By adding sin⁡60∘\sin 60^\circ and sin⁡30∘\sin 30^\circ, we get 3+12\frac{\sqrt{3} + 1}{2}.

Question 17

Assertion (A): cot⁡60∘=13\cot 60^\circ = \frac{1}{\sqrt{3}}.
Reason (R): cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \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: cot⁡60∘=cos⁡60∘sin⁡60∘=1/23/2=13\cot 60^\circ = \frac{\cos 60^\circ}{\sin 60^\circ} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}.

Question 18

Assertion (A): sec⁡45∘=2\sec 45^\circ = \sqrt{2}.
Reason (R): cos⁡45∘=12\cos 45^\circ = \frac{1}{\sqrt{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: Since sec⁡45∘=1cos⁡45∘=2\sec 45^\circ = \frac{1}{\cos 45^\circ} = \sqrt{2}.

Question 19

Assertion (A): cos⁡90∘=0\cos 90^\circ = 0.
Reason (R): The cosine function at 90° equals the x-coordinate of the unit circle point at this 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: A) Both A and R are true, and R is the correct explanation of A.
  • Explanation: At 90°, the point on the unit circle has an x-coordinate of 0, so cos⁡90∘=0\cos 90^\circ = 0.

Question 20

Assertion (A): sin⁡(90∘−θ)=cos⁡θ\sin(90^\circ - \theta) = \cos \theta.
Reason (R): Sine and cosine of complementary angles are equal.

  • 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: This is an identity for complementary angles.

Question 21

Assertion (A): tan⁡30∘⋅cot⁡30∘=1\tan 30^\circ \cdot \cot 30^\circ = 1.
Reason (R): tan⁡30∘=13\tan 30^\circ = \frac{1}{\sqrt{3}} and cot⁡30∘=3\cot 30^\circ = \sqrt{3}.

  • 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∘⋅cot⁡30∘=1\tan 30^\circ \cdot \cot 30^\circ = 1.

Question 22

Assertion (A): sin⁡45∘×cos⁡45∘=12\sin 45^\circ \times \cos 45^\circ = \frac{1}{2}.
Reason (R): sin⁡45∘=cos⁡45∘=12\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{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⁡45∘×cos⁡45∘=12×12=12\sin 45^\circ \times \cos 45^\circ = \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}} = \frac{1}{2}.

Question 23

Assertion (A): The value of tan⁡60∘\tan 60^\circ is equal to 3\sqrt{3}.
Reason (R): tan⁡60∘\tan 60^\circ can be derived from the ratio of sin⁡60∘\sin 60^\circ and cos⁡60∘\cos 60^\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: tan⁡60∘=sin⁡60∘cos⁡60∘=3/21/2=3\tan 60^\circ = \frac{\sin 60^\circ}{\cos 60^\circ} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}.

Question 24

Assertion (A): The value of sin⁡(90∘−30∘)\sin(90^\circ - 30^\circ) is equal to cos⁡30∘\cos 30^\circ.
Reason (R): The sine of a complementary angle is equal to the cosine of the angle itself.

  • 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⁡(90∘−30∘)=sin⁡60∘=cos⁡30∘\sin(90^\circ - 30^\circ) = \sin 60^\circ = \cos 30^\circ.

Question 25

Assertion (A): csc⁡45∘=2\csc 45^\circ = \sqrt{2}.
Reason (R): Cosecant is the reciprocal of sine, and sin⁡45∘=12\sin 45^\circ = \frac{1}{\sqrt{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: Since csc⁡45∘=1sin⁡45∘=2\csc 45^\circ = \frac{1}{\sin 45^\circ} = \sqrt{2}.