Graphs of Trigonometric Functions

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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