To find the exact value of $\cot(315^\circ)$ using reference angles, follow these steps:

Step 1: Find the reference angle

The given angle is $315^\circ$, which is in the fourth quadrant. The reference angle is the positive acute angle formed with the x-axis.

To calculate the reference angle, subtract $315^\circ$ from $360^\circ$: $360^\circ - 315^\circ = 45^\circ$ So, the reference angle is $45^\circ$.

Step 2: Determine the cotangent of the reference angle

The cotangent of $45^\circ$ is known: $\cot(45^\circ) = 1$

Step 3: Determine the sign of $\cot(315^\circ)$

In the fourth quadrant, the cotangent function is negative. Therefore: $\cot(315^\circ) = -\cot(45^\circ) = -1$

Final Answer:

$\cot(315^\circ) = -1$

Would you like further explanation on any step?

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Here are 5 related questions:

1. What is the reference angle for $225^\circ$?
2. How do you find the exact value of $\cot(120^\circ)$?
3. What is the value of $\cot(30^\circ)$?
4. How do you find the reference angle for angles greater than $360^\circ$?
5. What is the difference between cotangent and tangent?

Tip: Cotangent is the reciprocal of tangent. If $\tan(\theta) = x$, then $\cot(\theta) = \frac{1}{x}$.