Question

Sketch graphs of $\sin x\mathrm{and}\cos x\mathrm{on}\left[-2\mathrm{\pi },2\mathrm{\pi }\right].$

(a) Use the graph of $\sin x$to determine where $\sin x$is increasing and where it is decreasing.

(b) Use the graph of $\cos x$to determine where $\cos x$is positive and where it is negative.

(c) Explain why your answers to parts (a) and (b) suggest that $\cos x$is the derivative of $\sin x.$

Short answer

(a) The graph of $\sin x$increases at $\frac{-\mathrm{\pi }}{2}\mathrm{to}\frac{-3\mathrm{\pi }}{2},0\mathrm{to}\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\mathrm{to}2\mathrm{\pi }$and decreases at localid="1648770535664" $\frac{\mathrm{\pi }}{2}\mathrm{to}\frac{3\mathrm{\pi }}{2},0\mathrm{to}\frac{-\mathrm{\pi }}{2},\frac{-3\mathrm{\pi }}{2}\mathrm{to}-2\mathrm{\pi }.$

(b) The graph of $\cos x$is positive at $\frac{-\mathrm{\pi }}{2}\mathrm{to}\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\mathrm{to}2\mathrm{\pi },\frac{-3\mathrm{\pi }}{2}\mathrm{to}-2\mathrm{\pi }$and negative at $\frac{\mathrm{\pi }}{2}\mathrm{to}\frac{3\mathrm{\pi }}{2},\frac{-3\mathrm{\pi }}{2}\mathrm{to}\frac{-\mathrm{\pi }}{2}.$

(c) Intervals where the graph of $\sin x$increases and decreases are opposite of $\cos x,$so $\cos x$might be derivative of$\sin x.$

Step-by-step solution

Step 1. Given information.  

The given trigonometric Functions are$\sin x\mathrm{and}\cos x\mathrm{on}\left[-2\mathrm{\pi },2\mathrm{\pi }\right].$

Step 2. Graph of trigonometric functions.

Plot the graph of $\sin x\mathrm{on}\left[-2\mathrm{\pi },2\mathrm{\pi }\right].$

The graph of $\sin x$states that the function increases at $\frac{-\mathrm{\pi }}{2}\mathrm{to}\frac{-3\mathrm{\pi }}{2},0\mathrm{to}\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\mathrm{to}2\mathrm{\pi }.$

The graph of $\sin x$states that the function decreases at$\frac{\mathrm{\pi }}{2}\mathrm{to}\frac{3\mathrm{\pi }}{2},0\mathrm{to}\frac{-\mathrm{\pi }}{2},\frac{-3\mathrm{\pi }}{2}\mathrm{to}-2\mathrm{\pi }.$

Step 3. Part (b)

Plot the graph of $\cos x\mathrm{on}\left[-2\mathrm{\pi },2\mathrm{\pi }\right].$

The graph of $\cos x$states that the function is positive at$\frac{-\mathrm{\pi }}{2}\mathrm{to}\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\mathrm{to}2\mathrm{\pi },\frac{-3\mathrm{\pi }}{2}\mathrm{to}-2\mathrm{\pi }.$

The graph of $\cos x$states that the function is negative at$\frac{\mathrm{\pi }}{2}\mathrm{to}\frac{3\mathrm{\pi }}{2},\frac{-3\mathrm{\pi }}{2}\mathrm{to}\frac{-\mathrm{\pi }}{2}.$

Step 4. Part (c)

plot the graph of$\sin x\mathrm{and}\cos x\mathrm{on}\left[-2\mathrm{\pi },2\mathrm{\pi }\right].$

Intervals where the graph of $\sin x$increases and decreases are opposite of$\cos x.$

so $\cos x$might be derivative of $\sin x.$