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 $13$.

Example 4:Fun Trig Optimization!

Your goal is to find the maximum value of the following expression: $\sin x+\cos x+\sin x\cos x$.

First, we let $u=\sin x+\cos x$. Then $u^2=1+2\sin x\cos x$, so $\sin x\cos x=\frac{u^2-1}{2}$. The expression becomes $u+\frac{u^2-1}{2}$ with $-\sqrt{2}\le u\le\sqrt{2}$. Since $u+\frac{u^2-1}{2}$ is increasing on this interval, the maximum occurs at $u=\sqrt{2}$ and equals $\sqrt{2}+\frac{1}{2}$.

Example 5: Interesting Trig Telescoping Product!

Your job is to evaluate the following expression: $\prod_{k=1}^{n}\left(1+\tan^2\frac{\pi}{2^{k+2}}\right)$.

We can use $1+\tan^2\theta=\sec^2\theta$ and $\prod_{k=1}^{n}\cos\frac{x}{2^k}=\frac{\sin x}{2^n\sin(x/2^n)}$, and so the product simplifies to $2^{2n+1}\sin^2\left(\frac{\pi}{2^{n+2}}\right)$.

Example 6: Trigonometric Equation!

Your goal is to solve $\sin x+\sin3x+\sin5x=0$.

We can use $\sin x+\sin5x=2\sin3x\cos2x$, from which we obtain $\sin3x(1+2\cos2x)=0$, which gives $x=\frac{k\pi}{3}$ for all integers $k$.

Example 7: Interesting Trig Expression

Your job is to evaluate the following expression: $\sin20^\circ\sin40^\circ\sin80^\circ$.

We can use product-to-sum identities, which gives us: $\sin20^\circ\sin40^\circ\sin80^\circ=\frac{\sqrt{3}}{8}$.

Example 8: Law of Sines

In triangle $ABC$, $A=30^\circ$, $B=45^\circ$, and $a+b=1$. Your job is to find $c$.

By the Law of Sines, $c=\frac{2\sqrt{3}+2-\sqrt{6}-\sqrt{2}}{2}$.

Example 9: Product Identity

Your job is to find $\prod_{k=1}^{4}\cos\left(\frac{2^k\pi}{31}\right)$.

Using $\sin(2^n x)=2^n\sin x\prod_{j=0}^{n-1}\cos(2^j x)$ and $\sin\left(\frac{32\pi}{31}\right)=-\sin\left(\frac{\pi}{31}\right)$, the product equals $-\frac{1}{32\cos(\pi/31)}$.

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