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$Find the following trigonometric values. Express your answers exactly. {:[cos(150^(@))=],[sin(150^(@))=]:}$

Find the following trigonometric values. $\newline$ Express your answers exactly. $\newline$ $\begin{array}{l} \cos \left(150^{\circ}\right)= \\ \sin \left(150^{\circ}\right)= \end{array}$

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Evaluate an exponential function

Full solution

Q. Find the following trigonometric values.

\newline

Express your answers exactly.

\newline

\begin{array}{l} \cos \left(150^{\circ}\right)= \\ \sin \left(150^{\circ}\right)= \end{array}

Unit Circle Approach: To find the exact values of $\cos(150°)$ and $\sin(150°)$ , we can use the unit circle and the fact that $150°$ is in the second quadrant where cosine is negative and sine is positive. The reference angle for $150°$ is $180° - 150° = 30°$ . We can use the known values for $\cos(30°)$ and $\sin(30°)$ to find the values for $\cos(150°)$ and $\sin(150°)$ .
Cosine of $150$ °: The exact value of $\cos(30°)$ is $\sqrt{3}/2$ and since cosine is negative in the second quadrant, $\cos(150°) = -\sqrt{3}/2$ .
Sine of $150$ °: The exact value of $\sin(30°)$ is $\frac{1}{2}$ and since sine is positive in the second quadrant, $\sin(150°) = \frac{1}{2}$ .

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