Math 10C · Trigonometry
Three ratios, one triangle
The whole of this unit is: label the sides relative to the angle you care about, work out which two you have, and pick the ratio that contains them. Everything else is calculator work.
- 1. The words, first
- 2. The method, every time
- 3. Worked examples
- 4. Why the ratios work at all
- 5. What costs marks
The words, first
The idea: Opposite and adjacent are relative to an angle. The hypotenuse is not — and keeping that straight is most of the unit.
| Word | What it means |
|---|---|
| Hypotenuse | The side opposite the right angle, and always the longest. It does not change when you switch which acute angle you are working from. |
| Opposite | The side across from the angle you are using. It does change when you switch angles. |
| Adjacent | The side next to that angle that is not the hypotenuse. |
| Ratio | A comparison of two lengths. Because the ratio depends only on the angle, one table of values serves every right triangle of that shape. |
| SOH CAH TOA | Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. |
| Inverse trig function | sin⁻¹, cos⁻¹, tan⁻¹. They take a ratio and give back an angle — the opposite direction from the ordinary functions. |
| Angle of elevation | Measured upward from the horizontal. The angle of depression is measured downward, and for the same line of sight the two are equal. |
The method, every time
The idea: Draw it, label it, choose the ratio that has your two knowns and your one unknown.
- Draw the triangle and mark the right angle.
- Mark the angle you are working from, and label the three sides from that angle: hypotenuse, opposite, adjacent.
- Tick what you know and circle what you want. Two of the three sides, or one side and an angle.
- Choose the ratio that mentions exactly those. If you have opposite and adjacent, it is tangent, and the hypotenuse never enters it.
- Solve, then check it is sensible. The side opposite the bigger angle must be longer.
Step two is the one people skip, and it is the one that prevents the errors.
Worked examples
The idea: Finding a side uses a trig function; finding an angle uses its inverse.
Finding a side. An angle of 35° with the adjacent side 8 cm. Opposite and adjacent means tangent:
tan 35° = x / 8 → x = 8 tan 35° ≈ 5.6 cm
Check: 35° is less than 45°, so the opposite side should be shorter than the adjacent one — and it is.
Finding an angle. A 6 m ladder reaches 5 m up a wall. The height is opposite the angle and the ladder is the hypotenuse:
sin θ = 5/6 → θ = sin⁻¹(0.833) ≈ 56°
A height problem. From 40 m away, the angle of elevation to the top of a tower is 32°:
tan 32° = h / 40 → h ≈ 25 m
Why the ratios work at all
The idea: Two triangles with the same angles are the same shape at different sizes, so their corresponding sides are in the same proportion. The ratio therefore belongs to the angle, not to the triangle.
Take a 35° right triangle and double every side. The sides are all twice as long, but opposite ÷ adjacent is unchanged, because both parts doubled. That is why a single value of tan 35° works for every triangle containing a 35° angle, however large.
This is also why trigonometry is useful in the first place: measure an angle and one distance you can reach, and you can calculate a distance you cannot — the height of a tower, the width of a river, the depth of a canyon.
What costs marks
The idea: One of these is a calculator setting and it is silent.
- Radian mode. Everything comes out wrong and nothing looks obviously wrong. Check the display before you start.
- Labelling the sides once and reusing them for the other angle. Opposite and adjacent swap.
- Using sin instead of sin⁻¹ when finding an angle, or the reverse.
- Reporting the wrong angle. Check which side is opposite the one you found.