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Where Is the Dot?

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Watch the film below.

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Imagine the dot starts at the point $(1,0)$ and turns anticlockwise.

Estimate the height of the dot above the horizontal axis after it has turned through $45^\circ$.

Estimate the angle that the dot needs to turn in order to be exactly $0.5$ units above the horizontal axis.

Show how you can use Pythagoras' Theorem to calculate the height of the dot above the horizontal axis after it has turned through $45^\circ$.

Again, without resorting to Trigonometry, calculate the height of the dot above the horizontal axis after it has turned through $30^\circ$ and $60^\circ$?

Are there any other angles for which you can calculate the height of the dot above the horizontal axis?