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$Find the following trigonometric values. Express your answers exactly. {:[cos(315^(@))=],[sin(315^(@))=]:}$

Find the following trigonometric values. $\newline$ Express your answers exactly. $\newline$ $\begin{array}{l} \cos \left(315^{\circ}\right)= \\ \sin \left(315^{\circ}\right)= \end{array}$

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Algebra 2

Evaluate exponential functions

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Q. Find the following trigonometric values.

\newline

Express your answers exactly.

\newline

\begin{array}{l} \cos \left(315^{\circ}\right)= \\ \sin \left(315^{\circ}\right)= \end{array}

Using the unit circle: To find the exact values of $\cos(315^\circ)$ and $\sin(315^\circ)$ , we can use the unit circle and the properties of cosine and sine for angles in standard position. The angle of $315^\circ$ is located in the fourth quadrant, where cosine is positive and sine is negative. We can also recognize that $315^\circ$ is the same as $360^\circ - 45^\circ$ , which means it is the reference angle of $45^\circ$ in the fourth quadrant.
Recognizing the reference angle: The cosine and sine of $45^\circ$ are known values. Since $45^\circ$ is a special angle, we know that $\cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}$ . In the fourth quadrant, cosine remains positive, but sine becomes negative.
Exact values for $\cos(315^\circ)$ and $\sin(315^\circ)$ : Now we can write the exact values for $\cos(315^\circ)$ and $\sin(315^\circ)$ using the reference angle of $45^\circ$ . For $\cos(315^\circ)$ , since cosine is positive in the fourth quadrant, it will be the same as $\cos(45^\circ)$ , which is $\sqrt{2}/2$ . For $\sin(315^\circ)$ , since sine is negative in the fourth quadrant, it will be the negative of $\sin(45^\circ)$ , which is $\sin(315^\circ)$ $0$ .
Exact values for $\cos(315^\circ)$ and $\sin(315^\circ)$ : Now we can write the exact values for $\cos(315^\circ)$ and $\sin(315^\circ)$ using the reference angle of $45^\circ$ . For $\cos(315^\circ)$ , since cosine is positive in the fourth quadrant, it will be the same as $\cos(45^\circ)$ , which is $\sqrt{2}/2$ . For $\sin(315^\circ)$ , since sine is negative in the fourth quadrant, it will be the negative of $\sin(45^\circ)$ , which is $\sin(315^\circ)$ $0$ .Therefore, the exact values are: $\newline$ $\sin(315^\circ)$ $1$ $\newline$ $\sin(315^\circ)$ $2$ $\newline$ These are the exact values for the cosine and sine of $\sin(315^\circ)$ $3$ .