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$-90^(@) <= theta <= 90^(@). Find the value of theta in degrees. {:[sin(theta)=-(sqrt3)/(2)],[theta=◻]:} Submit$

$-90^{\circ} \leq \theta \leq 90^{\circ}$ . Find the value of $\theta$ in degrees. $\newline$ $\begin{array}{l} \sin (\theta)=-\frac{\sqrt{3}}{2} \\ \theta=\square \end{array}$ $\newline$ Submit

Full solution

-90^{\circ} \leq \theta \leq 90^{\circ}

. Find the value of

\theta

in degrees.

\newline

\begin{array}{l} \sin (\theta)=-\frac{\sqrt{3}}{2} \\ \theta=\square \end{array}

\newline

Submit

Identify Theta Values: Identify the values of $\theta$ for which the sine function equals negative square root of $3$ divided by $2$ . The sine function equals negative square root of $3$ divided by $2$ at specific reference angles in the unit circle. Since the value is negative, we are looking for angles in the third or fourth quadrant. However, since $\theta$ is restricted to the range $[-90$ degrees, $90$ degrees $]$ , we are only interested in the fourth quadrant where sine is negative.
Determine Reference Angle: Determine the reference angle whose sine is $\sqrt{3} / 2$ . The reference angle for which the sine function equals $\sqrt{3} / 2$ is $60$ degrees. This is because $\sin(60^\circ) = \sqrt{3}/2$ .
Find Fourth Quadrant Angle: Find the angle in the fourth quadrant that corresponds to the reference angle. $\newline$ Since the sine is negative and we are looking for an angle in the fourth quadrant, the angle $\theta$ will be $-60$ degrees because it is the negative of the reference angle.
Check Validity: Check if the found angle is within the given range. $\newline$ The angle $-60$ degrees is within the range $[-90$ degrees, $90$ degrees $]$ , so it is a valid solution.