Find the following trigonometric values.
Express your answers exactly.
\[\cos(315^\circ)=\]
\[\sin(315^\circ)=\]

Let's determine the trigonometric values for the given angles.

### 1. \( \cos(315^\circ) \)

First, note that \( 315^\circ \) is in the fourth quadrant, where cosine is positive and sine is negative.

To simplify, we find the reference angle. The reference angle is calculated as:

\[
360^\circ - 315^\circ = 45^\circ
\]

Since \( \cos(315^\circ) \) corresponds to \( \cos(45^\circ) \) in the fourth quadrant:

\[
\cos(315^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}
\]

### 2. \( \sin(315^\circ) \)

Again, using the reference angle of \( 45^\circ \) and knowing sine is negative in the fourth quadrant:

\[
\sin(315^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}
\]

### Final Answers

\[
\cos(315^\circ) = \frac{\sqrt{2}}{2}
\]

\[
\sin(315^\circ) = -\frac{\sqrt{2}}{2}
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the reference angle for \( 210^\circ \)?
2. What is \( \cos(120^\circ) \) expressed exactly?
3. How do you determine the quadrant of an angle?
4. What is the exact value of \( \tan(315^\circ) \)?
5. How do reference angles work in trigonometry?

**Tip:** Remember that trigonometric functions change signs depending on the quadrant of the angle.

Learn how to find the trigonometric values of cosine and sine for the angle 315 degrees using reference angles and trigonometric rules. Detailed step-by-step solutions provided.

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