Overview

Trig in contests often appears in geometry or algebra. Memorize the core identities and recognize when to apply the Law of Sines or Cosines.

Key Ideas

• $\sin^2\theta+\cos^2\theta=1$.
• Law of Cosines: $c^2=a^2+b^2-2ab\cos C$.
• Law of Sines: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.

Core Skills

Choose the Right Tool

Use angle identities to simplify expressions, and the triangle laws for geometry data with sides/angles.

Build Special Angles

Express $15^\circ, 75^\circ, 105^\circ$ as sums/differences of $30^\circ,45^\circ,60^\circ$.

Use Auxiliary Angles

Rewrite $a\sin x+b\cos x$ as $R\sin(x+\phi)$ to get extrema quickly.

Worked Example

Find $\cos 60^\circ$. Using the unit circle or a $30-60-90$ triangle, $\cos 60^\circ=\frac{1}{2}$.

More Examples

Example 1: Law of Sines

In triangle $ABC$, $A=30^\circ$, $B=45^\circ$, and $a=10$. Find $b$.

$b=10\sqrt{2}$.

Example 2: Angle Addition

Compute $\sin 15^\circ$.

$\sin(45^\circ-30^\circ)=\frac{\sqrt{6}-\sqrt{2}}{4}$.

Example 3: Auxiliary Angle

Find the maximum of $5\sin x + 12\cos x$.

$R=13$, so the maximum is $13$.

Strategy Checklist

• Identify whether the problem is algebraic or geometric.
• Use triangle laws when side-angle pairs are known.
• Use special-angle decomposition to keep radicals exact.

Common Pitfalls

• Mixing degrees and radians.
• Applying Law of Sines to SAS data.
• Forgetting to check quadrant signs.

Practice Problems