Question

find the equations of the tangent lines to the curve y = sin x at x = -π, -π/2, and 0. 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?

Explanation:

Step1: Find the derivative of $y = \sin x$

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

Step2: Evaluate the derivative at $x =-\pi$

When $x =-\pi$, $y'=\cos(-\pi)=- 1$.

Step3: Find the $y$-coordinate at $x =-\pi$

When $x =-\pi$, $y=\sin(-\pi)=0$.

Step4: Use the point - slope form $y - y_1=m(x - x_1)$

Here $m=-1$, $x_1 =-\pi$ and $y_1 = 0$. Substituting into the point - slope form $y-0=-1(x + \pi)$.

Answer:

$y=-x-\pi$