# 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/