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Chapter 8: Introduction to Trigonometry

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This page provides comprehensive Chapter 8: Introduction to Trigonometry - Assertion Reason Worksheet - SJMaths. Assertion and Reason type questions for Class 10 Introduction to Trigonometry. Practice for CBSE Board Exams.

Assertion-Reason Worksheet

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Directions:

In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:

  • (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • (B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
  • (C) Assertion (A) is true but Reason (R) is false.
  • (D) Assertion (A) is false but Reason (R) is true.
  1. Question 1:
    Assertion (A): $\sin^2 60^\circ + \cos^2 60^\circ = 1$.
    Reason (R): For any angle $\theta$, $\sin^2 \theta + \cos^2 \theta = 1$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: A is true as $(\frac{\sqrt{3}}{2})^2 + (\frac{1}{2})^2 = \frac{3}{4} + \frac{1}{4} = 1$.
    Step 2: R is the fundamental trigonometric identity. R explains A.
  2. Question 2:
    Assertion (A): The value of $\tan 45^\circ$ is less than $\cos 60^\circ$.
    Reason (R): $\tan 45^\circ = 1$ and $\cos 60^\circ = \frac{1}{2}$.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Step 1: $\tan 45^\circ = 1$ and $\cos 60^\circ = 0.5$.
    Step 2: $1 < 0.5$ is false. So A is false.
    Step 3: R is true.
  3. Question 3:
    Assertion (A): $\sec A = \frac{12}{5}$ for some value of angle A.
    Reason (R): The value of $\sec A$ is always $\ge 1$ or $\le -1$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: $\frac{12}{5} = 2.4$, which is $> 1$. So A is true.
    Step 2: R correctly states the range of secant function. R explains A.
  4. Question 4:
    Assertion (A): If $\cos A = \frac{4}{5}$, then $\tan A = \frac{3}{4}$.
    Reason (R): $\tan A = \frac{\sin A}{\cos A}$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: $\sin A = \sqrt{1 - (4/5)^2} = \sqrt{9/25} = 3/5$.
    Step 2: $\tan A = \frac{3/5}{4/5} = \frac{3}{4}$. A is true.
    Step 3: R is the formula used to find $\tan A$.
  5. Question 5:
    Assertion (A): The value of $2\sin \theta$ can be 3.
    Reason (R): The value of $\sin \theta$ lies between -1 and 1.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Step 1: If $2\sin \theta = 3$, then $\sin \theta = 1.5$.
    Step 2: Since max value of $\sin \theta$ is 1, A is false.
    Step 3: R is true.
  6. Question 6:
    Assertion (A): $\sin A \cos A \tan A + \cos A \sin A \cot A = 1$.
    Reason (R): $\tan A \cdot \cot A = 1$.
    (A)
    (B)
    (C)
    (D)
    Solution: (B)
    Step 1: LHS $= \sin A \cos A (\frac{\sin A}{\cos A}) + \cos A \sin A (\frac{\cos A}{\sin A}) = \sin^2 A + \cos^2 A = 1$. A is true.
    Step 2: R is true, but the main reason for A is the identity $\sin^2 A + \cos^2 A = 1$. So B is correct.
  7. Question 7:
    Assertion (A): If $4 \tan \theta = 3$, then $\frac{4 \sin \theta - \cos \theta}{4 \sin \theta + \cos \theta} = \frac{1}{2}$.
    Reason (R): $\tan \theta = \frac{\sin \theta}{\cos \theta}$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: Divide numerator and denominator by $\cos \theta$: $\frac{4\tan \theta - 1}{4\tan \theta + 1}$.
    Step 2: Substitute $4\tan \theta = 3$: $\frac{3-1}{3+1} = \frac{2}{4} = \frac{1}{2}$. A is true.
    Step 3: R is the property used to convert sin/cos to tan.
  8. Question 8:
    Assertion (A): $(1+\tan^2 A)(1-\sin A)(1+\sin A) = 1$.
    Reason (R): $\sec^2 A - \tan^2 A = 1$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: $(1+\tan^2 A) = \sec^2 A$. $(1-\sin A)(1+\sin A) = 1-\sin^2 A = \cos^2 A$.
    Step 2: $\sec^2 A \cdot \cos^2 A = 1$. A is true.
    Step 3: R is the identity used for the first term.
  9. Question 9:
    Assertion (A): In a right triangle, the hypotenuse is the longest side.
    Reason (R): The side opposite to the largest angle in a triangle is the longest side.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: In a right triangle, the right angle ($90^\circ$) is the largest angle.
    Step 2: Therefore, the side opposite to it (hypotenuse) is the longest. R explains A.
  10. Question 10:
    Assertion (A): $\sin(A+B) = \sin A + \sin B$.
    Reason (R): For any value of $\theta$, $\sin \theta$ is unique.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Step 1: Let $A=30^\circ, B=60^\circ$. $\sin(90^\circ) = 1$. $\sin 30^\circ + \sin 60^\circ = 0.5 + 0.866 \neq 1$. A is false.
    Step 2: R is true (function definition).
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