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.