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Question 20

The value of

\sin 75^\circ - \sin 165^\circ

is ....

$-1$

$-\frac{1}{2}\sqrt{2}$

$\frac{1}{2}\sqrt{2}$

$\sqrt{3}$

$\sqrt{2}$

To solve $\sin 75^\circ - \sin 165^\circ$ , use the sine difference formula

\sin A - \sin B = 2\cos\frac{1}{2}(A + B)\sin\frac{1}{2}(A - B)

Substitute $A = 75^\circ$ and $B = 165^\circ$

\sin 75^\circ - \sin 165^\circ = 2\cos\frac{1}{2}(75^\circ + 165^\circ)\sin\frac{1}{2}(75^\circ - 165^\circ)

= 2\cos 120^\circ \sin(-45^\circ)

Use the identities $\cos 120^\circ = \cos(180^\circ - 60^\circ) = -\cos 60^\circ$ and $\sin(-45^\circ) = -\sin 45^\circ$

= 2 \cdot \cos(180^\circ - 60^\circ) \cdot (-\sin 45^\circ)

= 2 \cdot (-\cos 60^\circ) \cdot (-\sin 45^\circ)

= 2\cos 60^\circ \sin 45^\circ

Substitute values $\cos 60^\circ = \frac{1}{2}$ and $\sin 45^\circ = \frac{1}{2}\sqrt{2}$

= 2 \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\sqrt{2}\right)

= 1 \cdot \frac{1}{2}\sqrt{2}

= \frac{1}{2}\sqrt{2}

Therefore, the value of $\sin 75^\circ - \sin 165^\circ = \frac{1}{2}\sqrt{2}$

Exercises