# Graphs of Trigonometric Functions - Problem Solving

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

## Problem Solving - Basic

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

Solution: Note that by graphing, we can see that \( \cos^2(\sin(x)) = \cos^2(\sin(x+ k\pi)) \) holds true for all integer \(k\). Thus the fundamental period of \(f(x) \) is \(\pi\).P.S: We can also solve this via compound angle formula.

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

Solution: Because \(y = \sin(x) \) has a fundamental period of 360 degrees, then \(\sin((10^{10})^\circ) = \sin((10^{10} \bmod{360})^\circ = \sin(280^\circ) = -\sin(80^\circ) \). 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, \(\tan(\cot(1^\circ)) \) or \(\tan(\tan(1^\circ)) \)?

Solution: Because \(\cot(1^\circ) \) is the reciprocal of \(\tan(1^\circ) \) which is rather small, than \( \tan(1^\circ) < \cot(1^\circ) \). And because both of them are positive and less than 90, then 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

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

Above shows the value of \(A,B,C\) and \(D\). Which of these answer choices is true?

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 \(f_k(f_k(\alpha))=\alpha,\) but \(f_k(\alpha )\neq \alpha\)?

**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/