Addition formulas: practice problems

The addition formulas

Example: $\sin 75^\circ = \sin(45^\circ + 30^\circ) = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\cdot\frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}$.

The acute angle $\theta$ between lines with slopes $m_1$ and $m_2$ satisfies $\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1m_2}\right|$.

Double- and half-angle formulas

For equations like $\cos 2\theta + \sin\theta = 0$, use $\cos 2\theta = 1 - 2\sin^2\theta$ to get a quadratic in $\sin\theta$. Choose the version of the formula that keeps the function you want.

The harmonic form

Plot the point $(a,\ b)$: its distance from the origin is $\sqrt{a^2 + b^2}$ and its angle with the positive $x$-axis is $\alpha$. Example: $\sin\theta + \sqrt{3}\cos\theta = 2\sin\left(\theta + \frac{\pi}{3}\right)$.

This shows that $a\sin\theta + b\cos\theta$ lies between $-\sqrt{a^2 + b^2}$ and $\sqrt{a^2 + b^2}$. If $\theta$ is restricted, track the range of $\theta + \alpha$.

Substitution for max and min

Put $t = \sin\theta + \cos\theta$. Squaring gives $\sin\theta\cos\theta = \frac{t^2 - 1}{2}$, and the harmonic form gives $-\sqrt{2} \leq t \leq \sqrt{2}$. An expression such as $\sin\theta + \cos\theta + \sin\theta\cos\theta$ becomes a quadratic in $t$.