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The problem in the image asks for the equations of the tangent lines to the curve \( y = \cos(x) \) at several points \( x = -\frac{\pi}{2}, 0, \text{and} \frac{\pi}{2} \), and to graph the curve over a specific interval along with its tangent lines.

### Let's focus on part of the problem that asks:
"What is the equation of the tangent line to the curve at \( x = -\frac{\pi}{2} \)?"

### Steps to solve:
1. **Find the derivative of the function** \( y = \cos(x) \). This gives the slope of the tangent line.
   \[
   \frac{dy}{dx} = -\sin(x)
   \]

2. **Evaluate the derivative at** \( x = -\frac{\pi}{2} \) to get the slope of the tangent line:
   \[
   \frac{dy}{dx}\bigg|_{x=-\frac{\pi}{2}} = -\sin\left(-\frac{\pi}{2}\right) = -(-1) = 1
   \]
   So, the slope of the tangent line is \( 1 \).

3. **Find the y-coordinate at** \( x = -\frac{\pi}{2} \):
   \[
   y = \cos\left(-\frac{\pi}{2}\right) = 0
   \]
   So, the point of tangency is \( \left(-\frac{\pi}{2}, 0\right) \).

4. **Use the point-slope form of the equation of a line**:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \( m = 1 \) is the slope, and \( \left(x_1, y_1\right) = \left(-\frac{\pi}{2}, 0\right) \) is the point of tangency.
   Substituting the values:
   \[
   y - 0 = 1\left(x - \left(-\frac{\pi}{2}\right)\right)
   \]
   \[
   y = x + \frac{\pi}{2}
   \]

Thus, the equation of the tangent line at \( x = -\frac{\pi}{2} \) is:
\[
y = x + \frac{\pi}{2}
\]

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Would you like more details on graphing the curve and the tangent lines or explanations for the other points?

Here are some related questions for deeper understanding:
1. How do we determine the slope of a tangent line from the derivative?
2. What is the significance of evaluating the function at a particular point?
3. How do you find the equation of a tangent line if the function is more complex?
4. Can the process be generalized for other trigonometric functions like \( y = \sin(x) \)?
5. What are the possible uses of tangent lines in applied problems?

**Tip:** The point-slope form of a line is very useful for quickly finding the equation of a tangent line once the slope is known.

Find the equations of the tangent lines to the curve y = cos(x) at x = -π/2, 0, and π/2. Graph the curve over the interval [-3π/2, 2π] together with its tangent lines. Label the curve and each tangent line.

Learn how to find the equations of the tangent lines to the curve y = cos(x) at x = -π/2, 0, and π/2. This problem involves the use of calculus concepts such as derivatives and point-slope form of a line. A graph of the curve over the interval [-3π/2, 2π] with labeled tangent lines is also required. Suitable for high school students studying calculus.

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