Right-triangle trigonometry
Choose a ratio from the reference angle and calculate exact side lengths and areas.
What you will learn
- Label opposite, adjacent and hypotenuse correctly.
- Use exact 30°, 45° and 60° relationships.
The reference angle determines the labels
The hypotenuse is opposite the right angle and is always the longest side. The opposite and adjacent labels depend on which acute angle you are using. Mark that angle first, then choose sine, cosine or tangent according to the sides known and required.
When finding area, use two perpendicular lengths as base and height. The hypotenuse is not automatically a perpendicular height.
Keep exact values exact
A 30°–60°–90° triangle has side ratios 1:√3:2, with the side 1 opposite 30°. A 45°–45°–90° triangle has ratios 1:1:√2. These let you calculate without a calculator.
Simplify surds where possible and leave them symbolic when exact form is requested. You can check a result using Pythagoras: the squares of the perpendicular sides must add to the square of the hypotenuse.
Worked example
A right triangle has hypotenuse 10 cm and an acute angle of 30°. Find its exact area.
Show the worked solution
- The side opposite 30° is 10×1/2=5 cm.
- The adjacent side is 10×√3/2=5√3 cm.
- The perpendicular sides give area 1/2×5×5√3.
Answer 25√3/2 cm²
Common mistakes
- Opposite and adjacent change when the reference angle changes.
- Using a decimal for √3 loses exactness.
What can you explain now?
A right triangle has two equal perpendicular sides of length 6 cm. Find the hypotenuse.
Compare with the explanation
6√2 cm
Pythagoras gives c²=36+36=72, so c=√72=6√2.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.