To solve the equation \(\tan(C) = -1\) for \(C\), we need to determine the angles \(C\) where the tangent function equals \(-1\).

### Solution:

The tangent function is periodic with a period of \(\pi\) radians (or 180°). The tangent of an angle is \(-1\) at the following general angles:

\[
C = \frac{3\pi}{4} + n\pi \quad \text{or} \quad C = 135^\circ + 180^\circ n
\]

where \(n\) is any integer.

So, the general solution for \(C\) can be expressed as:

\[
C = \frac{3\pi}{4} + n\pi \quad \text{radians}
\]
or
\[
C = 135^\circ + 180^\circ n
\]

where \(n\) is any integer.

### Would you like more details on any part of this solution? Here are some additional questions that might help you deepen your understanding:

1. How is the tangent function defined in terms of sine and cosine?
2. Why is the period of the tangent function \(\pi\) instead of \(2\pi\) like sine and cosine?
3. What are the exact values of \(\tan(C)\) at common angles like \(0^\circ\), \(45^\circ\), \(90^\circ\), and \(180^\circ\)?
4. How can you graphically represent the tangent function to visualize where it equals \(-1\)?
5. What are the implications of the periodicity of the tangent function on solving trigonometric equations?

#### Tip:
When solving trigonometric equations, always consider the periodic nature of the trigonometric functions, as it will help you identify all possible solutions.

Learn how to solve the equation tan(C) = -1 with detailed explanations and examples. Explore the periodic nature of the tangent function and its solutions. Suitable for students in Grades 10-12.

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