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Let's work through the problem step by step.

### Problem:
We need to find the equations of the tangent lines to the curve \( y = \cos(x) \) at three points: \( x = -\pi \), \( x = -\frac{\pi}{2} \), and \( x = \frac{\pi}{2} \).

The general equation of the tangent line to a curve at a point \( x = a \) is:

\[
y = f'(a)(x - a) + f(a)
\]

Where:
- \( f'(a) \) is the slope of the tangent line at \( x = a \) (which we get from the derivative of \( y = \cos(x) \)),
- \( f(a) \) is the value of the function \( y = \cos(x) \) at \( x = a \).

### 1. Find the derivative of \( y = \cos(x) \):
The derivative of \( \cos(x) \) is:

\[
f'(x) = -\sin(x)
\]

### 2. Calculate the slopes at each point:

#### At \( x = -\pi \):
\[
f'(-\pi) = -\sin(-\pi) = 0
\]
The slope of the tangent line is 0, so the line is horizontal.

The value of \( f(-\pi) = \cos(-\pi) = -1 \).

The equation of the tangent line is:
\[
y = -1
\]

#### At \( x = -\frac{\pi}{2} \):
\[
f'\left(-\frac{\pi}{2}\right) = -\sin\left(-\frac{\pi}{2}\right) = 1
\]
The slope is 1.

The value of \( f\left(-\frac{\pi}{2}\right) = \cos\left(-\frac{\pi}{2}\right) = 0 \).

The equation of the tangent line is:
\[
y = 1\left(x + \frac{\pi}{2}\right) = x + \frac{\pi}{2}
\]

#### At \( x = \frac{\pi}{2} \):
\[
f'\left(\frac{\pi}{2}\right) = -\sin\left(\frac{\pi}{2}\right) = -1
\]
The slope is -1.

The value of \( f\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \).

The equation of the tangent line is:
\[
y = -1\left(x - \frac{\pi}{2}\right) = -x + \frac{\pi}{2}
\]

### Summary of the Tangent Line Equations:
- At \( x = -\pi \): \( y = -1 \)
- At \( x = -\frac{\pi}{2} \): \( y = x + \frac{\pi}{2} \)
- At \( x = \frac{\pi}{2} \): \( y = -x + \frac{\pi}{2} \)

If you need more details or graphing explanations, let me know!

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Here are five related questions:
1. How is the derivative of \( \cos(x) \) derived, and what does it represent graphically?
2. What happens to the tangent line when the slope is zero?
3. How can we interpret the symmetry of the cosine function in relation to its tangent lines?
4. Can you find the equation of a tangent line for \( y = \sin(x) \) at a specific point?
5. How do we graphically determine the tangent line to a curve without calculating the derivative?

**Tip:** When finding the tangent line to a curve, always begin by calculating the derivative of the function to determine the slope at the desired point.

Find the equations of the tangent lines to the curve y = cos(x) at x = -π, -π/2, and π/2. Graph the curve over the interval [-3π/2, 2π] together with its tangent lines. Label the curve and each tangent line. What is the equation of the tangent line (l) to the curve at x = -π?

This problem involves finding the equations of tangent lines to the curve y = cos(x) at x = -π, -π/2, and π/2. Explore how to compute the slope of tangent lines using differentiation of trigonometric functions and graph the curve with its tangent lines.

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