Introduction to Trigonometry (SOH-CAH-TOA)

What is trigonometry?

Trigonometry is the study of the relationship between the angles and the side lengths of triangles. In this lesson we focus on right-angled triangles — triangles with one 90° angle. The amazing idea is this: in a right triangle, if you know one angle and one side, you can work out every other side and angle. That lets you measure things you could never reach with a ruler, like the height of a tree or the distance to a mountain.

This topic pairs naturally with the Pythagorean theorem, which deals with the sides of a right triangle. Trigonometry adds the angles into the picture.

Naming the three sides

Before anything else, you must label the sides relative to the angle you care about. Pick the angle you are working with (often labelled θ, the Greek letter "theta"). Then:

• The hypotenuse is the longest side, always opposite the right angle. Its name never changes.
• The opposite side is the one directly across from your chosen angle θ.
• The adjacent side is the one next to θ (but not the hypotenuse).

So "opposite" and "adjacent" depend on which angle you picked, while the hypotenuse is fixed.

The three trig ratios: SOH-CAH-TOA

In any right triangle, the ratio of two particular sides depends only on the angle, not on the size of the triangle. There are three key ratios, remembered by the famous phrase SOH-CAH-TOA:

Ratio	Formula	Memory part
Sine	sin θ = Opposite / Hypotenuse	SOH
Cosine	cos θ = Adjacent / Hypotenuse	CAH
Tangent	tan θ = Opposite / Adjacent	TOA

Say it out loud: "SOH-CAH-TOA" → Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj. Memorising this single phrase unlocks the whole topic.

Finding a trig ratio

Worked example: A right triangle has, for angle θ, an opposite side of 3 cm, an adjacent side of 4 cm, and a hypotenuse of 5 cm. Find sin θ, cos θ, and tan θ.

1. sin θ = opp / hyp = 3 / 5 = 0.6.
2. cos θ = adj / hyp = 4 / 5 = 0.8.
3. tan θ = opp / adj = 3 / 4 = 0.75.

That is all there is to forming a ratio — pick the right two sides and divide.

Finding a missing side

If you know an angle and one side, you can find another side. The plan is: choose the ratio that links the side you know with the side you want, then solve.

Worked example: In a right triangle, one angle is 30° and the hypotenuse is 10 cm. Find the opposite side.

1. We know the hypotenuse and want the opposite, so use the ratio with both: sine (SOH).
2. Write it: sin 30° = opposite / 10.
3. From a calculator (in degree mode), sin 30° = 0.5.
4. So 0.5 = opposite / 10.
5. Multiply both sides by 10: opposite = 0.5 × 10 = 5 cm.

Worked example: An angle is 40° and the adjacent side is 6 m. Find the hypotenuse.

1. We know the adjacent and want the hypotenuse → use cosine (CAH).
2. cos 40° = adjacent / hypotenuse = 6 / hypotenuse.
3. cos 40° ≈ 0.766.
4. So 0.766 = 6 / hypotenuse. Rearrange: hypotenuse = 6 / 0.766 ≈ 7.83 m.

Worked example: An angle is 35° and its adjacent side is 8 cm. Find the opposite side.

1. Adjacent known, opposite wanted → use tangent (TOA).
2. tan 35° = opposite / 8.
3. tan 35° ≈ 0.700.
4. opposite = 0.700 × 8 = 5.6 cm.

Finding a missing angle

If you know two sides, you can find the angle using the inverse trig functions: sin⁻¹, cos⁻¹, tan⁻¹ (on a calculator usually the "shift" or "2nd" key before sin/cos/tan). An inverse function takes a ratio and gives back the angle that produced it.

Worked example: The opposite side is 5 and the hypotenuse is 10. Find the angle θ.

1. We have opposite and hypotenuse → use sine.
2. sin θ = opp / hyp = 5 / 10 = 0.5.
3. To undo the sine, take the inverse: θ = sin⁻¹(0.5).
4. From the calculator, θ = 30°.

Worked example: The opposite side is 7 and the adjacent side is 7. Find θ.

1. Opposite and adjacent → use tangent.
2. tan θ = 7 / 7 = 1.
3. θ = tan⁻¹(1) = 45°.

When opposite equals adjacent, the angle is always 45° — a tidy result worth remembering.

A real-world example

Problem: You stand 20 m from the base of a tower. Looking up to the top, the angle of elevation is 50°. How tall is the tower (ignoring your eye height)?

1. Draw the right triangle: the 20 m is the adjacent side (along the ground), and the tower height is the opposite side. The angle is 50°.
2. Adjacent known, opposite wanted → tangent: tan 50° = height / 20.
3. tan 50° ≈ 1.19.
4. height = 1.19 × 20 ≈ 23.8 m.

You just measured a tower's height with a tape measure and one angle. That is the power of trigonometry.

Common mistakes to avoid

• Wrong calculator mode. Set it to DEGREES for these problems. If sin 30° does not give 0.5, your calculator is in radians.
• Mislabelling sides. Always identify opposite and adjacent relative to the angle you are using, not the right angle.
• Forgetting to rearrange. If the unknown is on the bottom of the fraction (like in cos 40° = 6 / hyp), you must divide, not multiply.

Practice activity

Use SOH-CAH-TOA. Keep your calculator in degree mode.

1. A triangle has opposite = 6, hypotenuse = 12. Find sin θ and then θ.
2. An angle is 60° and the hypotenuse is 14. Find the opposite side. (sin 60° ≈ 0.866)
3. An angle is 25° and the adjacent side is 9. Find the opposite side. (tan 25° ≈ 0.466)
4. Opposite = 4, adjacent = 4. Find θ.

Answers:

1. sin θ = 6/12 = 0.5, so θ = sin⁻¹(0.5) = 30°.
2. sin 60° = opp/14, so opp = 0.866 × 14 ≈ 12.1.
3. tan 25° = opp/9, so opp = 0.466 × 9 ≈ 4.19.
4. tan θ = 4/4 = 1, so θ = tan⁻¹(1) = 45°.

Summary

Trigonometry links the angles and sides of a right triangle through three ratios captured in SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj. To find a missing side, choose the ratio that connects the known side to the wanted side and solve. To find a missing angle, form the ratio from two sides and apply the inverse function. With these tools you can calculate heights, distances and slopes that are impossible to measure directly.