To find the equation of the tangent line to the curve $y = \cos(x)$ at the point $x = \frac{\pi}{2}$, follow these steps:

Step 1: Find the slope of the tangent line

The slope of the tangent line is given by the derivative of the function $y = \cos(x)$.

$y' = \frac{d}{dx} \left( \cos(x) \right) = -\sin(x)$

Now, evaluate the derivative at $x = \frac{\pi}{2}$:

$y'\left( \frac{\pi}{2} \right) = -\sin\left( \frac{\pi}{2} \right) = -1$

So, the slope of the tangent line at $x = \frac{\pi}{2}$ is $-1$.

Step 2: Find the point of tangency

We already know that $x = \frac{\pi}{2}$. To find the corresponding $y$-value:

$y = \cos\left( \frac{\pi}{2} \right) = 0$

Thus, the point of tangency is $\left( \frac{\pi}{2}, 0 \right)$.

Step 3: Use the point-slope form to find the equation of the tangent line

The point-slope form of a line is:

$y - y_1 = m(x - x_1)$

Substitute the slope $m = -1$ and the point $\left( \frac{\pi}{2}, 0 \right)$:

$y - 0 = -1 \left( x - \frac{\pi}{2} \right)$

Simplify the equation:

$y = -x + \frac{\pi}{2}$

Final Answer:

The equation of the tangent line to the curve $y = \cos(x)$ at $x = \frac{\pi}{2}$ is:

$y = -x + \frac{\pi}{2}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How would you find the tangent to $y = \sin(x)$ at $x = 0$?
2. Can you find the tangent to $y = \cos(x)$ at $x = 0$?
3. What is the normal line equation at $x = \frac{\pi}{2}$?
4. How does the slope of the tangent change as $x$ approaches 0 from $x = \frac{\pi}{2}$?
5. How would you find the tangent to a parametric curve at a given point?

Tip: When finding tangents, always differentiate first to obtain the slope at the desired point.