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Trigonometry: Compound Angles

How can we work with trigonometric functions of sums and differences?

1
How can expressions such as $\sin (x+y)$ be expanded?
2
What are the double-angle formulae?
3
What is a geometrical interpretation of expressions such as $\cos(x+30^{\circ})$?
4
What do graphs such as $y=3 \cos x + 4 \sin x$ look like?
5
How can products such as $\sin x \cos 3x$ be written as a sum or difference of trigonometric functions?

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