Guide · For JSS 3 – SS 2 students · 7 min read
Trigonometry without fear: SOH CAH TOA and beyond
Name the sides, pick the right ratio with SOH CAH TOA, find missing sides and angles, solve elevation and depression questions, learn the special angles and meet the sine and cosine rules.
By the MathPlug team, Reflex System · Published
Trigonometry has a reputation for being hard, but most of it rests on one simple fact: in a right-angled triangle, the angles decide the shape, and the shape decides how the sides compare. Sine, cosine and tangent are just names for those comparisons.
If you can name the three sides of a right-angled triangle and remember SOH CAH TOA, you can already answer a large share of trigonometry questions. This guide builds from there, one step at a time, to angles of elevation and depression, the special angles and the sine and cosine rules.
Step 1: name the sides
Every right-angled triangle has three sides, and their names depend on which angle you are working with. Call that angle θ (the Greek letter theta).
- Hypotenuse (H): the longest side, opposite the right angle. It never changes.
- Opposite (O): the side directly across from θ. It does not touch θ.
- Adjacent (A): the side next to θ that is not the hypotenuse.
💡 Tip: Label the sides H, O and A on your diagram before you do anything else. Find the hypotenuse first, then the opposite. The one left over is the adjacent. If you change to the other acute angle, O and A swap places.
Step 2: SOH CAH TOA
The three trigonometric ratios compare pairs of sides:
| Memory word | Ratio | Uses |
|---|---|---|
| SOH | sin θ = OppositeHypotenuse | O and H |
| CAH | cos θ = AdjacentHypotenuse | A and H |
| TOA | tan θ = OppositeAdjacent | O and A |
To choose the ratio, look at the two sides involved: the one you know and the one you want (or, when finding an angle, the two you know). Then pick the ratio that uses exactly those two sides.
Before any calculation, make sure your calculator is in DEG (degrees) mode. A quick test: sin 30° should give 0.5. If you get about −0.988, your calculator is in radians and every answer will be wrong.
Finding a missing side
In triangle PQR, angle Q = 90°, angle P = 35° and the hypotenuse PR = 12 cm. Find QR, the side opposite angle P. Give your answer to 3 significant figures.
- Label the sides from angle P: PR = H = 12 cm, QR = O (wanted).
- O and H are involved, so use SOH: sin 35° = QR12
- Multiply both sides by 12: QR = 12 × sin 35°
- sin 35° = 0.5736 (to 4 d.p.), so QR = 12 × 0.5736 = 6.883…
- Answer: QR = 6.88 cm (3 s.f.)
- Check: the opposite side must be shorter than the hypotenuse, and 6.88 < 12. Using CAH, PQ = 12 × cos 35° = 9.830…, and 6.8832 + 9.8302 ≈ 47.37 + 96.63 = 144 = 122, so Pythagoras agrees.
If the unknown side is on the bottom of the fraction, for example tan 40° = 8x, rearrange to x = 8tan 40°. Students often multiply here instead of dividing; a quick check that the answer is a sensible length will catch it.
Finding a missing angle
When you know two sides and want the angle, find the ratio first, then use the inverse function on your calculator: sin−1, cos−1 or tan−1 (usually SHIFT then sin, cos or tan). It answers the question "which angle has this ratio?"
A right-angled triangle has the side opposite angle θ equal to 5 cm and the side adjacent to θ equal to 8 cm. Find θ to 1 decimal place.
- O = 5 cm and A = 8 cm, so use TOA: tan θ = 58 = 0.625
- θ = tan−1(0.625)
- θ = 32.005…°
- Answer: θ = 32.0° (1 d.p.)
- Check: tan 32.0° = 0.6249 (to 4 d.p.), which is very close to 0.625. And the angle is less than 45°, which fits, because the opposite side is shorter than the adjacent.
Angles of elevation and depression
The angle of elevation is the angle you look up through, measured from the horizontal. The angle of depression is the angle you look down through, also measured from the horizontal. Always draw the horizontal line first, then the line of sight.
Tunde stands 30 m from the foot of the school flagpole on level ground. From his eye, 1.5 m above the ground, the angle of elevation of the top of the pole is 40°. Find the height of the flagpole to 3 significant figures.
- Draw a right-angled triangle from Tunde's eye: horizontal side 30 m (adjacent to 40°), vertical side h (opposite).
- O and A are involved, so use TOA: tan 40° = h30
- h = 30 × tan 40° = 30 × 0.8391 = 25.17… m
- This is the height above Tunde's eye, so add his eye height: 25.17… + 1.5 = 26.67… m
- Answer: the flagpole is 26.7 m tall (3 s.f.)
From the top of a 50 m telecom mast, Ada's brother sees a car parked on level ground. The angle of depression of the car is 25°. How far is the car from the foot of the mast?
- The angle of depression from the top equals the angle of elevation from the car (alternate angles), so the angle at the car is 25°.
- From the car: mast height 50 m is opposite, distance d is adjacent. Use TOA: tan 25° = 50d
- d = 50tan 25° = 500.4663 = 107.2… m
- Answer: the car is 107 m from the foot of the mast (3 s.f.)
The special angles: 30°, 45° and 60°
Some exam questions say "without using a calculator or tables". For these you need exact values of the special angles. They come from an equilateral triangle cut in half (30° and 60°) and a right-angled isosceles triangle (45°).
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 12 | √32 | 1√3 = √33 |
| 45° | √22 | √22 | 1 |
| 60° | √32 | 12 | √3 |
Notice the pattern: the sin column read downwards matches the cos column read upwards. So sin 30° = cos 60° and sin 60° = cos 30°. The special angles formula card has a diagram of both triangles.
Beyond right angles: the sine and cosine rules
SOH CAH TOA only works in right-angled triangles. For any other triangle, label the sides a, b and c opposite the angles A, B and C, and use one of two rules:
- Sine rule: asin A = bsin B = csin C. Use it when you know a side and the angle opposite it.
- Cosine rule: a2 = b2 + c2 − 2bc cos A. Use it when you know two sides and the angle between them, or all three sides.
(a) In triangle ABC, A = 40°, B = 65° and a = 8 cm. Find b. (b) In another triangle, b = 5 cm, c = 7 cm and A = 60°. Find a.
- (a) Sine rule: bsin 65° = 8sin 40°, so b = 8 × sin 65°sin 40° = 8 × 0.90630.6428 = 11.28… cm
- Answer: b = 11.3 cm (3 s.f.)
- (b) Cosine rule: a2 = 52 + 72 − 2 × 5 × 7 × cos 60° = 25 + 49 − 70 × 0.5 = 74 − 35 = 39
- a = √39 = 6.244… cm
- Answer: a = 6.24 cm (3 s.f.)
💡 Tip: Keep the full calculator value until the last line and round only once at the end: lengths to 3 significant figures and angles to 1 decimal place, unless the question says otherwise. Rounding too early can change your final answer.
Questions students ask
- What does SOH CAH TOA stand for?
- Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. It is a memory aid for the three trigonometric ratios in a right-angled triangle.
- How do you know whether to use sin, cos or tan?
- Label the sides H, O and A from the angle you are using. Find the two sides involved (the one you know and the one you want), then choose the ratio that uses exactly those two.
- Why is my calculator giving wrong trigonometry answers?
- It is probably in radian (RAD) or gradian (GRA) mode. Switch it to degree (DEG) mode and test it: sin 30° should give exactly 0.5.
- When do you use the sine rule instead of SOH CAH TOA?
- Use SOH CAH TOA only in right-angled triangles. In any other triangle, use the sine rule when you know a side and its opposite angle, and the cosine rule when you know two sides and the angle between them, or all three sides.
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