Angles and sides
Trigonometry connects the angles of a triangle to the lengths of its sides. It starts with the right triangle.
The Pythagorean theorem relates three sides. Trigonometry adds angles, so you can find a side from an angle, or an angle from two sides.
Labelling sides
Label sides relative to the angle you’re using (call it θ, "theta"):
| Side | Where |
|---|---|
| Hypotenuse | across from the 90° angle (always the longest) |
| Opposite side | across from θ |
| Adjacent side | touching θ, but not the hypotenuse |
If you switch to the other acute angle, opposite and adjacent swap.
The three ratios
| Ratio | Definition |
|---|---|
| sin θ | opposite / hypotenuse |
| cos θ | adjacent / hypotenuse |
| tan θ | opposite / adjacent |
Memory aid: SOH CAH TOA.
Why ratios work
All right triangles with the same angle θ are similar, so their side ratios are the same no matter how big the triangle is. That’s why a calculator can give you sin 30° = 0.5 for any triangle.
Finding a side
Example: θ = 35°, hypotenuse = 10. Find the opposite side.
- sin 35° = opp/10
- opp = 10 sin 35° ≈ 10 × 0.574 = 5.74
Example: θ = 50°, adjacent = 8. Find the opposite side.
- tan 50° = opp/8
- opp = 8 tan 50° ≈ 9.53
Finding an angle
Use the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹).
Example: opposite = 7, adjacent = 12
- tan θ = 7/12
- θ = tan⁻¹(7/12) ≈ 30.3°
Solving the whole triangle
To "solve" a right triangle means to find all sides and angles. Use:
- Angle sum: the two acute angles add to 90°
- Pythagoras for the third side
- Trig ratios for anything else
Calculator tip
Make sure the calculator is in degree mode (DEG). In radian mode, sin 30 gives −0.988 instead of 0.5.
Worked examples
Example 1: In a right triangle with legs 3 and 4, find sin of the angle opposite the 3. Hypotenuse = 5 → sin = 3/5.
Example 2: Hypotenuse 20, angle 60°. Adjacent side? 20 cos 60° = 10.
Example 3: Opposite 5, hypotenuse 13. Angle? sin⁻¹(5/13) ≈ 22.6°.
Common mistakes
- Mislabelling opposite and adjacent.
- Calculator in radian mode.
- Dividing instead of multiplying (or the reverse) when isolating a side.
Try it yourself
- Write tan θ for a triangle with opposite 9 and adjacent 12.
- θ = 40°, hypotenuse 15. Find the opposite side.
- θ = 25°, adjacent 10. Find the hypotenuse.
- Opposite 8, adjacent 8. Find θ.
Answers: 1) 9/12 = 0.75 2) ≈ 9.64 3) ≈ 11.03 4) 45°
Quick recap
- Label the hypotenuse, opposite side and adjacent side from the angle.
- SOH CAH TOA gives sin, cos and tan.
- Use inverse trig to find angles; keep the calculator in degrees.