Overview

Triangle problems often combine geometry and trig. The law of sines and law of cosines are the two standard tools.

Core Skills

Match Sides and Angles

Always pair side $a$ with angle $A$ and so on. Most errors are mismatches.

Choose the Right Law

Use law of sines for AAS/ASA and circumradius questions; use law of cosines for SAS/SSS or when a triangle is not right.

Use the Area Formula

If two sides and the included angle are known, $A=\tfrac12 ab\sin C$ is often faster than solving for a third side.

Law of Sines

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R.$

Use it for AAS/ASA or when a circumradius is involved.

Law of Cosines

$a^2 = b^2 + c^2 - 2bc\cos A.$

Use it for SAS or SSS in a non-right triangle.

Area with Trig

$A = \frac{1}{2}ab\sin C.$

Worked Example

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

$\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}$ gives $b=10\sqrt{2}$.

More Examples

Example 1: Law of Cosines

Triangle has sides $b=7$, $c=9$ and included angle $A=60^\circ$. Find $a$.

$a^2=7^2+9^2-2\cdot7\cdot9\cos 60^\circ=49+81-63=67$, so $a=\sqrt{67}$.

Example 2: Circumradius

If $a=12$ and $A=45^\circ$, find the circumradius $R$.

$a=2R\sin A$ gives $R=12/(\sqrt{2})=6\sqrt{2}$.

Example 3: Area

Two sides are $8$ and $10$ with included angle $30^\circ$. Find the area.

$A=\tfrac12\cdot 8\cdot 10\cdot \sin 30^\circ=20$.

Strategy Checklist

• Identify the known side-angle pairs.
• Choose law of sines or cosines based on the data.
• Use the area formula if two sides and included angle are given.

Common Pitfalls

• Mixing up sides opposite angles.
• Applying the law of sines to SAS data.
• Using degrees and radians inconsistently.

Practice Problems