find the tangent to y = cos x at x = pi/2

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.

Find the tangent to y = cos x at x = pi/2

Learn how to find the tangent line to the curve y = cos(x) at x = π/2. This step-by-step guide covers key calculus concepts, including differentiation and the point-slope form, suitable for high school students in Grades 10-12.

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