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Geometry

# Graphs of Trigonometric Functions: Level 3 Challenges

$\large \color{orangered}{f(x)=\sin^6x+\cos^6x+\sin^4x \cos^2x + \cos^4x \sin^2x}$ Find the fundamental period of the function $$\color{orangered}{f(x)}$$.

Over the entire real line, the number of values of x where the function

$f(x) = \cos x + \cos ( \sqrt{2}x )$ attains its maximum value is:

$\large \begin{cases}{y=\sin (x)} \\ {y=\cos (x)} \\ {y=\tan (x)} \\ {y=\csc (x)}\end{cases}$

The four graphs above are drawn on the same axes from $$x = 0$$ to $$x = \frac \pi2$$.

If a vertical line is drawn where the graphs of $$y=\cos (x)$$ and $$y = \tan (x)$$ intersect, this line intersects the graphs of $$y=\sin(x)$$ and $$y=\csc(x)$$ at points $$A$$ and $$B$$. What is the distance between $$A$$ and $$B$$?

How many real numbers $$x$$ satisfy $$\sin x = \frac{x}{100}?$$

What is the smallest positive integer $$n$$ such that $\large \displaystyle f(x)= \frac {\sin(\sin (nx))}{\tan \left(\frac{x}{n}\right) }$ has a fundamental period of $$6\pi?$$

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