Question

Find the value of $ \sin 60^\circ - \sin 80^\circ + \sin 100^\circ - \sin 120^\circ $

• A. \( 0 \)
• B. \( -1 \)
• C. \( 1 \)
• D. \( 2 \)

Hint:

Use the sum-to-product identities for sine and cosine to simplify trigonometric expressions involving subtraction of sines. This method often simplifies such problems significantly.

Answer 1

The Correct Option is A

We are asked to simplify the following expression: \[ \sin 60^\circ - \sin 80^\circ + \sin 100^\circ - \sin 120^\circ \] Using the sum-to-product identities for sine: \[ \sin A - \sin B = 2 \cos\left( \frac{A + B}{2} \right) \sin\left( \frac{A - B}{2} \right) \] First, simplify \( \sin 60^\circ - \sin 120^\circ \): \[ \sin 60^\circ - \sin 120^\circ = 2 \cos\left( \frac{60^\circ + 120^\circ}{2} \right) \sin\left( \frac{60^\circ - 120^\circ}{2} \right) \] \[ = 2 \cos(90^\circ) \sin(-30^\circ) = 0 \] Now simplify \( \sin 80^\circ - \sin 100^\circ \): \[ \sin 80^\circ - \sin 100^\circ = 2 \cos\left( \frac{80^\circ + 100^\circ}{2} \right) \sin\left( \frac{80^\circ - 100^\circ}{2} \right) \] \[ = 2 \cos(90^\circ) \sin(-10^\circ) = 0 \]
Thus, the entire expression simplifies to: \[ 0 + 0 = 0 \] Therefore, the value of the expression is 0.