Multiple choice

Consider the following statements : 1. If $3 \theta $ is an acute angle such that $\sin 3 \theta = \cos 2 \theta$, then the measurement of $\theta$ in radian equals to $\frac{\pi}{10}$. 2. One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to the diameter of that circle. Which of the above statements is/are correct ?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since 3θ is acute, sin 3θ = cos 2θ gives 3θ = 90° - 2θ, so 5θ = 90° and θ = π/10. One radian corresponds to an arc whose length equals the radius, not the diameter, so statement 2 is false.

AI explanation

Using the cofunction identity for acute angles, sin(3 theta) = cos(2 theta) means 3 theta + 2 theta = 90 degrees, so 5 theta = 90 degrees and theta = 18 degrees. Converting 18 degrees to radians gives 18 * (pi / 180) = pi / 10, making statement 1 correct. One radian is defined by an arc length equal to the radius, not the diameter, so statement 2 is incorrect. Therefore, only statement 1 is correct.

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