To find $\cot(-270^\circ)$, we first need to recall that:

$\cot(\theta) = \frac{1}{\tan(\theta)}$

Thus, we need to evaluate $\tan(-270^\circ)$ first.

Step 1: Understanding the angle

• $-270^\circ$ is a negative angle, so it is measured clockwise from the positive x-axis.
• Starting from $0^\circ$, going clockwise by 270° takes us to the negative y-axis. This corresponds to the angle $270^\circ$ but in the negative direction.

Step 2: Evaluating $\tan(-270^\circ)$

• The tangent of an angle is the ratio of the sine and cosine of that angle: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

For $\theta = -270^\circ$:

• $\sin(-270^\circ) = -1$
• $\cos(-270^\circ) = 0$

Since the cosine is 0, the tangent is undefined:

$\tan(-270^\circ) = \frac{-1}{0} \quad \text{(undefined)}$

Step 3: Conclusion for $\cot(-270^\circ)$

Since $\tan(-270^\circ)$ is undefined, $\cot(-270^\circ)$ is also undefined.

Let me know if you'd like further clarification or a more in-depth explanation!

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Related questions:

1. What is the cotangent of $45^\circ$?
2. How do we handle undefined values in trigonometric functions?
3. What is the general approach to finding cotangent values for negative angles?
4. Can you explain the relationship between sine, cosine, and tangent?
5. How do trigonometric functions behave in the different quadrants?

Tip: Remember that the cotangent function is undefined wherever the tangent function has a zero denominator, i.e., at multiples of $90^\circ$ (such as $\pm 270^\circ$, $\pm 90^\circ$, etc.).