1) Identify the triangle sides
For a right triangle, start by naming the sides relative to the given angle:
- Opposite: the side across from the angle
- Adjacent: the side next to the angle, not the hypotenuse
- Hypotenuse: the longest side, opposite the right angle
2) Match the ratio to the sides
Use the trig ratio that connects the sides you know and the side you need:
- Sine: (\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}})
- Cosine: (\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}})
- Tangent: (\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}})
A quick memory check is SOH–CAH–TOA.
3) Choose the correct function
Ask: which two side lengths are given or needed?
- If the problem uses opposite and hypotenuse, choose sine.
- If it uses adjacent and hypotenuse, choose cosine.
- If it uses opposite and adjacent, choose tangent.
4) Simplify the final answer
Write the ratio in simplest form. If a fraction can be reduced, reduce it. If a radical can be simplified, simplify it. Keep the expression exact when possible.
5) Check your choice
Make sure:
- the numerator and denominator match the correct sides,
- the angle reference is correct,
- the simplified answer still represents the same ratio.
If the answer does not use the right pair of sides, go back and re-label the triangle.