Graphs of Trigonometric Functions

Graphs of Trigonometric Functions: Level 3 Challenges


f(x)=sin6x+cos6x+sin4xcos2x+cos4xsin2x\large \color{orangered}{f(x)=\sin^6x+\cos^6x+\sin^4x \cos^2x + \cos^4x \sin^2x} Find the fundamental period of the function f(x)\color{orangered}{f(x)}.

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

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

{y=sin(x)y=cos(x)y=tan(x)y=csc(x) \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=0x = 0 to x=π2x = \frac \pi2 .

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

How many real numbers xx satisfy sinx=x100?\sin x = \frac{x}{100}?

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


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