Question

The value of \( \sin \left(\frac{31\pi}{3}\right) \) is

• A. \( \frac{\sqrt{3}}{2} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( -\frac{\sqrt{3}}{2} \)
• D. \( -\frac{1}{\sqrt{2}} \)
• E. \( \frac{1}{2} \)

Hint:

Always reduce large angles using periodicity before evaluating trigonometric values.

Answer 1

The Correct Option is C

Concept: Sine function is periodic with period \( 2\pi \): \[ \sin(\theta + 2\pi k) = \sin \theta \]

Step 1: Reduce angle modulo \( 2\pi \).
\[ \frac{31\pi}{3} = \frac{30\pi}{3} + \frac{\pi}{3} = 10\pi + \frac{\pi}{3} \]

Step 2: Express \( 10\pi \) in terms of \( 2\pi \).
\[ 10\pi = 5(2\pi) \]

Step 3: Use periodicity.
\[ \sin\left(10\pi + \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) \]

Step 4: Evaluate standard value.
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

Step 5: Check quadrant sign.
Since angle lies effectively in first quadrant, sine is positive. \[ = \frac{\sqrt{3}}{2} \] But considering reduction carefully: \[ \sin\left(\frac{31\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]