Question

Which of the following have the same value:
(a) \( \sin 120^\circ \)
(b) \( \cos 930^\circ \)
(c) \( \tan 840^\circ \)
(d) \( \cot(-1110^\circ) \)

• A. only (a) and (b)
• B. All (a), (b), (c), (d)
• C. only (a) and (c)
• D. only (c) and (d)

Hint:

When working with angles outside the standard range, always reduce them by subtracting or adding multiples of 360°.

Answer 1

The Correct Option is D

Step 1: Simplify each trigonometric function.
- For \( \sin 120^\circ \), \[ \sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} \] - For \( \cos 930^\circ \), \[ 930^\circ = 2 \times 360^\circ + 210^\circ \quad \Rightarrow \quad \cos 930^\circ = \cos 210^\circ = -\frac{\sqrt{3}}{2} \] - For \( \tan 840^\circ \), \[ 840^\circ = 2 \times 360^\circ + 120^\circ \quad \Rightarrow \quad \tan 840^\circ = \tan 120^\circ = -\sqrt{3} \] - For \( \cot(-1110^\circ) \), \[ -1110^\circ = -3 \times 360^\circ - 30^\circ \quad \Rightarrow \quad \cot(-1110^\circ) = \cot(-30^\circ) = \sqrt{3} \] Step 2: Identify the matching pairs.
We find that \[ \text{(C) } \tan 840^\circ = -\sqrt{3}, \quad \text{(D) } \cot(-1110^\circ) = \sqrt{3} \] Thus, only (C) and (D) have the same value.