Degree Ceremony
WLOG (without loss of generality) let the length of the hypotenuse be equal to 1.
Can you find two different expressions for $a$? Can you find two different expressions for $b$?
Can you find a relationship connecting $\sin^2(45^{\circ} +x)$ and $\sin^2(45^{\circ}-x)$?
You can draw the graph of the sine function. What about the graph of $y = \sin^2 x$? What about the periodicity of this graph?
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Telescoping Series
Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.
OK! Now Prove It
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
Overarch 2
Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?