Trigonometry
What is Trigonometry?
Trigonometry is the study of relationships between angles and sides in right-angled triangles. It allows you to find unknown sides or angles when you know some information about the triangle.
The three trigonometric ratios — sine, cosine, and tangent — are the core tools.
Labelling the Sides
Before using any trig ratio, correctly label the three sides relative to the angle θ you are working with:
- Hypotenuse (H): always opposite the right angle — the longest side of the triangle
- Opposite (O): the side directly opposite angle θ
- Adjacent (A): the side next to angle θ that is not the hypotenuse
The hypotenuse is always H. The opposite and adjacent depend on which angle you choose.
SOH CAH TOA
The memory aid SOH CAH TOA gives all three ratios:
SOH: sin θ = Opposite / Hypotenuse
CAH: cos θ = Adjacent / Hypotenuse
TOA: tan θ = Opposite / Adjacent
These ratios are fixed for any given angle in any right-angled triangle, regardless of the size of the triangle.
Finding a Missing Side
Step 1: Label H, O, A relative to the given angle.
Step 2: Identify which two sides are involved (one known, one unknown).
Step 3: Choose the ratio that uses both of those sides.
Step 4: Substitute and solve.
Example 1: Hypotenuse = 10cm, angle = 30°, find the opposite side.
sin 30° = O / H → 0.5 = O / 10 → O = 10 × 0.5 = 5cm
Example 2: A ladder 5m long leans against a wall at 60° to the ground. How high up the wall?
The wall height is opposite to 60°; the ladder is the hypotenuse.
sin 60° = O / 5 → 0.866 = O / 5 → O = 5 × 0.866 = 4.33m
Example 3: Angle = 40°, adjacent = 8cm, find the opposite.
tan 40° = O / A → O = 8 × tan 40° ≈ 8 × 0.839 = 6.71cm
Finding a Missing Angle
Use inverse trig functions (written sin⁻¹, cos⁻¹, tan⁻¹ — also called arcsin, arccos, arctan) to find an unknown angle.
Step 1: Identify which two sides are known.
Step 2: Choose the ratio connecting those sides.
Step 3: Apply the inverse function.
Example 1: Opposite = 7, adjacent = 7. Find θ.
tan θ = O/A = 7/7 = 1 → θ = tan⁻¹(1) = 45°
Example 2: Adjacent = 6cm, hypotenuse = 10cm. Find θ.
cos θ = A/H = 6/10 = 0.6 → θ = cos⁻¹(0.6) ≈ 53.1°
Which Ratio to Use?
| You know | You want | Use |
|---|---|---|
| H and θ | O | sin |
| H and θ | A | cos |
| A and θ | O | tan |
| O and θ | H | sin (rearrange) |
| O and A | θ | tan⁻¹ |
| O and H | θ | sin⁻¹ |
| A and H | θ | cos⁻¹ |
Key Trig Values
Memorise these exact values — they appear frequently in non-calculator questions:
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Note: √2/2 ≈ 0.707 and √3/2 ≈ 0.866.
Worked Example: Full Problem
A right-angled triangle has an angle of 35° and the side adjacent to it is 12cm. Find the hypotenuse and the opposite side.
Finding H: cos 35° = A/H → H = A / cos 35° = 12 / 0.819 ≈ 14.65cm
Finding O: tan 35° = O/A → O = 12 × tan 35° = 12 × 0.700 ≈ 8.40cm
Check: sin 35° = O/H = 8.40/14.65 ≈ 0.573 ≈ sin 35° ✓
Angles of Elevation and Depression
Angle of elevation: the angle measured upward from the horizontal to a point above.
Angle of depression: the angle measured downward from the horizontal to a point below.
These appear in real-world problems involving heights, distances, and line of sight.
Example: An observer 20m from the base of a building looks up at an angle of elevation of 50°. Find the height of the building.
tan 50° = height / 20 → height = 20 × tan 50° ≈ 20 × 1.192 = 23.84m
Common Mistakes
- Labelling O and A incorrectly — always label relative to the specific angle θ, not the right angle
- Using the wrong ratio — check twice: does SOH/CAH/TOA match the sides you have?
- Forgetting to use the inverse function when finding an angle (writing tan θ = 1.2 and leaving it there instead of applying tan⁻¹)
- Rounding too early in multi-step problems — keep full calculator precision until the final step
- Confusing degrees and radians on a calculator — ensure the calculator is set to degree mode
Tips and Tricks
- Write SOH CAH TOA at the top of every trig question
- Cover up the unknown in a triangle formed by S, O, H (or C, A, H or T, O, A) to see whether to multiply or divide
- For exact values: sin and cos of 30° and 60° are 1/2 and √3/2 — remember which is which by noting sin 30° = 0.5 (smaller angle, smaller value)
- tan 45° = 1 because at 45° the opposite and adjacent sides are equal
- Always sketch the triangle and label all known information before selecting a ratio