# Graphs of Trigonometric Functions - Problem Solving

To solve the problems on this page, you should be familiar with the following:

## Problem Solving - Basic

What is the fundamental period of the function $f(x) = \cos^2\big(\sin(x)\big)?$

Note that by graphing, we can see that $\cos^2\big(\sin(x)\big) = \cos^2\big(\sin(x+ k\pi)\big)$ holds true for all integer $k$. Thus the fundamental period of $f(x)$ is $\pi$. $_\square$

P.S.We can also solve this via compound angle formula.

Evaluate $\sin \big(10^{10}\big)^\circ \div \sin 80^\circ.$

Because $y = \sin x$ has a fundamental period of 360 degrees,

$\sin \big(10^{10}\big)^\circ = \sin \big(10^{10} \bmod{360}\big)^\circ = \sin 280^\circ = -\sin 80^\circ.$

Thus taking the quotient yields $-1$ as the answer. $_\square$

## Problem Solving - Intermediate

Given that $\tan 1^\circ > \frac1{90},$ which of these numbers is larger, $\tan(\tan 1^\circ)$ or $\tan(\cot 1^\circ)?$

Because $\cot 1^\circ$ is the reciprocal of $\tan 1^\circ,$ which is rather small, $\tan 1^\circ < \cot 1^\circ.$ Also, because both of them are positive and less than 90, they are in the first quadrant. With $y = \tan x$ as an increasing function, we have $\tan( \tan 1^\circ) < \tan(\cot 1^\circ).$ So the latter number is larger. $_\square$

## Problem Solving - Advanced

$A = \sin\left[\sin(1)\right] \\ B = \sin\left[\cos(1)\right] \\ C = \cos\left[\sin(1)\right] \\ D =\cos\left[\cos(1)\right]$

The above are the values of $A,B,C,$ and $D.$ Which of the answer choices is true?

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**Clarification:** All angles are measured in radians.

Inspiration.

For every integer $k,$ we define a function $f_k$ by the formula

$f_k(x)=100x-k\sin x.$

What is the smallest positive integer value of $k$ such that, for some real $\alpha$, we have $f_k\big(f_k(\alpha)\big)=\alpha,$ but $f_k(\alpha )\neq \alpha?$

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**Details and Assumptions:** The function is evaluated in radians. There is no degree symbol in the problem.

**Cite as:**Graphs of Trigonometric Functions - Problem Solving.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/graphs-of-trigonometric-functions/