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# Making Waves

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Age 16 to 18

Challenge Level

- Sketch the graph of $y=\sin(\cos x)$, where $x$ is in radians.

Here are some things to consider when sketching your graph:It might help to start by sketching $y=\cos x$. What are the maximum and minimum values of $\cos x$?

Where does the graph cross the $y$ axis? (You don't have to find a numerical value!)

Can the graph cross the $x$ axis? If so, where?

Is the function even, odd or neither? A function is even if $f(-x)=f(x)$ and it is odd if $f(-x) = -f(x)$. An even function has reflection symmetry over the $y$ axis and an odd function has rotational symmetry about the origin.

The functions $\sin x$ and $\cos x$ are periodic with period $2 \pi$. What about $\sin(\cos x)$?

- Sketch the graph of $y=\cos(\sin x)$.

- Show that $\cos(\sin x) > \sin(\cos x)$ when $x=0, \pi$ and $\frac{\pi} 2$.

Can you also show that this is true when $x=\frac{\pi} 4$?

- Show that $\cos(\sin x) > \sin(\cos x)$ for all values of $x$.

*Some possible methods for this last part are discussed in the Getting Started Section.*

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.