Overview

Contest trig often comes down to two ideas: rewrite angles into special angles and compress expressions like $a\sin x + b\cos x$.

Core Skills

Use Auxiliary Angles

Rewrite $a\sin x+b\cos x$ as $R\sin(x+\phi)$ or $R\cos(x-\phi)$ to read off maxima, minima, and phase shifts.

Build Special Angles

Express tricky angles as sums or differences of $30^\circ$, $45^\circ$, and $60^\circ$ to keep radicals exact.

Recognize Symmetry

If $\sin x$ and $\cos x$ appear symmetrically, try $\sin x=\cos(90^\circ-x)$.

Hidden Special Angles

Rewrite angles using $15^\circ$, $75^\circ$, $105^\circ$, or $165^\circ$ as sums/differences of $30^\circ$, $45^\circ$, $60^\circ$.

Example: $\cos 165^\circ = -\cos 15^\circ$.

Auxiliary Angle Method

For $a\sin x + b\cos x$, write:

$a\sin x + b\cos x = R\sin(x+\phi)$

where $R=\sqrt{a^2+b^2}$ and $\tan\phi=b/a$.

So the maximum is $R$ and the minimum is $-R$.

Worked Example

Find the maximum of $3\sin x + 4\cos x$.

$R=\sqrt{3^2+4^2}=5$, so the maximum is $5$.

More Examples

Example 1: Minimum Value

Find the minimum of $5\sin x - 12\cos x$.

$R=13$, so the minimum is $-13$.

Example 2: Exact Value

Compute $\cos 75^\circ$.

$\cos(45^\circ+30^\circ)=(\sqrt{6}-\sqrt{2})/4$.

Example 3: Phase Shift

Write $2\sin x+2\cos x$ in the form $R\sin(x+\phi)$.

$R=2\sqrt{2}$ and $\phi=45^\circ$.

Strategy Checklist

• Convert sums of sine and cosine to a single trig function.
• Use special-angle decompositions.
• Track degrees vs radians carefully.

Common Pitfalls

• Forgetting to factor $R$ in the auxiliary angle method.
• Using degrees in a radian-only context.
• Dropping the sign when converting to $R\sin(x+\phi)$.

Practice Problems