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 (i) to the curve at x = -π? y = -x - π what is the equation of the tangent line (ii) to the curve at x = -π/2? y = -1 what is the equation of the tangent line (iii) to the curve at x = 0?

Explanation:

Step1: Recall the formula for tangent - line

The equation of a tangent line to the curve $y = f(x)$ at $x = a$ is $y - f(a)=f^{\prime}(a)(x - a)$, where $f^{\prime}(x)$ is the derivative of $f(x)$. The derivative of $y=\sin x$ is $y^{\prime}=\cos x$.

Step2: Find $f(a)$ and $f^{\prime}(a)$ for $x = 0$

When $x = 0$, $f(0)=\sin(0)=0$, and $f^{\prime}(0)=\cos(0)=1$.

Step3: Substitute into the tangent - line formula

Substitute $a = 0$, $f(0)=0$, and $f^{\prime}(0)=1$ into $y - f(a)=f^{\prime}(a)(x - a)$. We get $y-0 = 1\times(x - 0)$, which simplifies to $y=x$.

Answer:

$y=x$