Question

find the equations of the tangent lines to the curve y = cos x at x = -\frac{\pi}{2},\frac{\pi}{2}, and \pi. graph the curve over the interval -\frac{3\pi}{2},2\pi together with its tangent lines. label the curve and each tangent line.
what is the equation of the tangent line (i) to the curve at x = -\frac{\pi}{2}?
y = x+\frac{\pi}{2}
what is the equation of the tangent line (ii) to the curve at x = \frac{\pi}{2}?

Explanation:

Step1: Recall derivative of cosine

The derivative of $y = \cos x$ is $y'=-\sin x$.

Step2: Find slope at $x = \frac{\pi}{2}$

Substitute $x=\frac{\pi}{2}$ into $y'$. So $m = -\sin(\frac{\pi}{2})=- 1$.

Step3: Find $y$-coordinate at $x = \frac{\pi}{2}$

Substitute $x = \frac{\pi}{2}$ into $y=\cos x$. We get $y=\cos(\frac{\pi}{2}) = 0$.

Step4: Use point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$. Here $x_1=\frac{\pi}{2}$, $y_1 = 0$ and $m=-1$. So $y-0=-1(x - \frac{\pi}{2})$, which simplifies to $y=-x+\frac{\pi}{2}$.

Answer:

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