Trigonometry: SOHCAHTOA

This lesson introduces right-angled triangle trigonometry for AQA GCSE Mathematics. You will learn how to label the sides of a right-angled triangle relative to a given angle, use the sine, cosine and tangent ratios to find missing sides and angles, and apply trigonometry in real-world problems. Trigonometry is a core Higher-tier topic but also appears on Foundation tier.

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Labelling the Sides

Before using trigonometry, you must label the three sides of a right-angled triangle relative to the angle you are working with (call it angle theta).

Side	Abbreviation	Position
Hypotenuse	Hyp	The longest side, opposite the right angle
Opposite	Opp	The side opposite the angle theta
Adjacent	Adj	The side next to the angle theta (not the hypotenuse)

graph TD
    A[Choose angle theta] --> B[Hypotenuse: opposite the right angle]
    A --> C[Opposite: opposite angle theta]
    A --> D[Adjacent: next to angle theta, not hypotenuse]

Exam Tip: The hypotenuse never changes — it is always opposite the right angle. But the "opposite" and "adjacent" sides depend on which angle you are working with. Always label relative to the angle in the question.

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The Three Trigonometric Ratios: SOHCAHTOA

SOHCAHTOA is the mnemonic for remembering the three ratios:

Ratio	Formula	Mnemonic
Sine	sin(theta) = Opposite / Hypotenuse	Some Old Horses
Cosine	cos(theta) = Adjacent / Hypotenuse	Can Always Hear
Tangent	tan(theta) = Opposite / Adjacent	Their Owners Approach

How to Choose the Right Ratio

1. Label the sides: Hyp, Opp, Adj (relative to the angle you are using).
2. Identify which two sides are involved (one known, one unknown).
3. Choose the ratio that uses those two sides.

Sides Involved	Use
Opp and Hyp	Sin
Adj and Hyp	Cos
Opp and Adj	Tan

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Finding a Missing Side

Worked Example 1: Finding the Opposite (using Sin)

In a right-angled triangle, the angle is 35 degrees and the hypotenuse is 12 cm. Find the opposite side.

Solution: sin(35) = Opp / 12 Opp = 12 x sin(35) Opp = 12 x 0.5736... Opp = 6.88 cm (2 d.p.)

Worked Example 2: Finding the Adjacent (using Cos)

In a right-angled triangle, the angle is 50 degrees and the hypotenuse is 20 cm. Find the adjacent side.

Solution: cos(50) = Adj / 20 Adj = 20 x cos(50) Adj = 20 x 0.6428... Adj = 12.86 cm (2 d.p.)

Worked Example 3: Finding the Opposite (using Tan)

In a right-angled triangle, the angle is 62 degrees and the adjacent side is 8 cm. Find the opposite side.

Solution: tan(62) = Opp / 8 Opp = 8 x tan(62) Opp = 8 x 1.8807... Opp = 15.05 cm (2 d.p.)

Worked Example 4: Finding the Hypotenuse

In a right-angled triangle, the angle is 28 degrees and the opposite side is 7 cm. Find the hypotenuse.

Solution: sin(28) = 7 / Hyp Hyp = 7 / sin(28) Hyp = 7 / 0.4695... Hyp = 14.91 cm (2 d.p.)

Exam Tip: When the unknown side is on the bottom of the fraction (the denominator), you need to rearrange. If sin(theta) = Opp / Hyp, then Hyp = Opp / sin(theta). Many students make errors here — practise the rearrangement.

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Finding a Missing Angle

To find a missing angle, use the inverse trigonometric functions: sin to the power of -1, cos to the power of -1, or tan to the power of -1 (written as arcsin, arccos, arctan on some calculators).

Worked Example 5

In a right-angled triangle, the opposite side is 6 cm and the hypotenuse is 10 cm. Find the angle.

Solution: sin(theta) = 6 / 10 = 0.6 theta = inverse sin(0.6) theta = 36.9 degrees (1 d.p.)

Worked Example 6

In a right-angled triangle, the adjacent side is 5 cm and the opposite side is 9 cm. Find the angle.

Solution: tan(theta) = 9 / 5 = 1.8 theta = inverse tan(1.8) theta = 60.9 degrees (1 d.p.)

Worked Example 7

In a right-angled triangle, the adjacent side is 8 cm and the hypotenuse is 11 cm. Find the angle.

Solution: cos(theta) = 8 / 11 = 0.7272... theta = inverse cos(0.7272...) theta = 43.3 degrees (1 d.p.)

Exam Tip: Make sure your calculator is in degree mode, not radian mode. If you get an unexpected answer (especially a very small decimal), check the mode setting. On most scientific calculators, look for "D" or "DEG" on the display.

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Applying Trigonometry in Context

Angles of Elevation and Depression

• Angle of elevation: the angle measured upward from the horizontal to a line of sight.
• Angle of depression: the angle measured downward from the horizontal to a line of sight.

Worked Example 8: Angle of Elevation

A person stands 50 m from the base of a building and looks up at the top. The angle of elevation is 32 degrees. Find the height of the building.

Solution: tan(32) = height / 50 height = 50 x tan(32) height = 50 x 0.6249... height = 31.2 m (1 d.p.)

Worked Example 9: Bearings and Trigonometry

A ship sails 15 km on a bearing of 060 degrees. How far east has it travelled?

Solution: The eastward distance is the side opposite to the angle measured from north. In the right-angled triangle formed: sin(60) = east / 15 east = 15 x sin(60) = 15 x 0.8660... = 13.0 km (1 d.p.)

Exam Tip: In bearing questions, always draw a diagram with a north line. The angle in the triangle is often not the bearing itself but needs to be calculated from it. Take care to identify which angle you are using.

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Exact Trigonometric Values [H]

You should memorise these exact values:

Angle	sin	cos	tan
0 degrees	0	1	0
30 degrees	1/2	(root 3)/2	1/(root 3) or (root 3)/3
45 degrees	(root 2)/2	(root 2)/2	1
60 degrees	(root 3)/2	1/2	root 3
90 degrees	1	0	undefined

Worked Example 10 [H]