$y=3^{\cos(x)}$ how to graph this trigonometric function

Question

Please help me with graphing this function $y=3^{\cos(x)}$ without graphing software.

Thanks in advance for all your procedures.

Answer 1

Some observations:

• $u \mapsto 3^u$ is order-preserving, so $3^{\cos x}$ is increasing (resp. decreasing, stationary) precisely when $\cos x$ is increasing (resp. decreasing, stationary).
• $\cos x$ is periodic with period $2\pi$, hence so is $3^{\cos x}$.
• $\cos x$ takes values between $-1$ and $1$, so $3^{\cos x}$ takes values between $\frac{1}{3}$ and $3$.

With this information, and by plugging in some known values of $x$ (say, multiples of $\frac{\pi}{4}$ or $\frac{\pi}{6}$ or something), you should be able to make a reasonably accurate plot.