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What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

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Loch Ness

Draw graphs of the sine and modulus functions and explain the humps.

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Squareness

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

Tangled Trig Graphs

Stage: 5 Challenge Level: Challenge Level:1
This anonymous solver correctly identified the remaining curves, and explained how to draw a graph of sin $x$ using the cosine function:

The red graph has equation $y=-\sin x$.
The green graph has equation $y =\sin 2x$.
The light blue graph has equation $y=-\sin 2x$.
The grey graph has equation $y=\sin 3x$.
The dark blue graph has equation $y=-\sin 3x$.

The graph $y=\cos x$ is the same shape as $y=\sin x$ but shifted along. I can make the shape of $y=\sin x$ by drawing the graph $y=\cos (x-90^{\circ})$.