Right-angled trigonometry | AQA GCSE Maths Notes

Visual model

Choose SOH, CAH or TOA from the sides named

\theta

adjacent to angleoppositehypotenuse

\text{SOH CAH TOA}

Gold-standard guide

20 mins

What you will learn

Use sine, cosine and tangent to find angles and lengths.

Use a clear step-by-step method for right-angled trigonometry.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier geometry and measures skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

SOH-CAH-TOA: sin(θ) = O/H, cos(θ) = A/H, tan(θ) = O/A

Step 1

Identify the known sides relative to the unknown angle

Opposite = 8.4 cm, Hypotenuse = 13.2 cm

Step 2

Write the trigonometric equation

sin(θ) = 8. $\frac{4}{1}3$ .2

Step 3

Apply the inverse sine

θ = sin^-1(8. $\frac{4}{1}3$ .2) ≈ sin^-1(0.6364) ≈ 39.5°

Watch out

Students label the opposite and adjacent sides incorrectly by confusing which angle they are working from

SOH

$sin(\theta ) = opposit\frac{e}{hypotenuse}.$

CAH

$cos(\theta ) = adjacen\frac{t}{hypotenuse}.$

TOA

$tan(\theta ) = opposit\frac{e}{adjacent}.$

Worked example

In a right-angled triangle, the opposite side is 8.4 cm and the hypotenuse is 13.2 cm. Find the angle between the hypotenuse and the adjacent side.

Identify the known sides relative to the unknown angle: Opposite = 8.4 cm, Hypotenuse = 13.2 cm. Use sine.

Write the trigonometric equation: sin(θ) = 8. $\frac{4}{1}3$ .2.

Apply the inverse sine: θ = sin^-1(8. $\frac{4}{1}3$ .2) ≈ sin^-1(0.6364) ≈ 39.5°.

Final answer

θ ≈ 39.5°

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

In a right-angled triangle, the opposite side is 8.4 cm and the hypotenuse is 13.2 cm. Find the angle between the hypotenuse and the adjacent side.

3 marks4 minsright-angled-trigonometry-worked

Show solution

Worked solution

1.Identify the known sides relative to the unknown angle: Opposite = 8.4 cm, Hypotenuse = 13.2 cm. Use sine.
2.Write the trigonometric equation: sin(θ) = 8. $\frac{4}{1}3$ .2.
3.Apply the inverse sine: θ = sin^-1(8. $\frac{4}{1}3$ .2) ≈ sin^-1(0.6364) ≈ 39.5°.

Final answer

θ ≈ 39.5°

Mark points

M1: identify the known sides relative to the unknown angle
M1: write the trigonometric equation
M1: apply the inverse sine
A1: θ ≈ 39.5°

Watch out

Students label the opposite and adjacent sides incorrectly by confusing which angle they are working from.Always label O, A and H relative to the angle marked with θ — not relative to any other angle in the triangle.

Diagnosticrecall

Find the side opposite a 35° angle in a right triangle with hypotenuse 20 cm.

1 mark2 minsright-angled-trigonometry-q1

Show solution

Worked solution

1.Spot the skill: SOH-CAH-TOA: sin(θ) = O/H, cos(θ) = A/H, tan(θ) = O/A.
2.Use the identify the known sides relative to the unknown angle stage first, then write the trigonometric equation.
3.Keep the final answer visible: ≈ 11.5 cm.

Final answer

≈ 11.5 cm

Mark points

M1: use the correct soh-cah-toa: sin(θ) = o/h, cos(θ) = a/h, tan(θ) = o/a. g. o = h × sin θ). g. θ = sin^-1(o/h)).label sides relative to the angle you are working with.
A1: ≈ 11.5 cm

Watch out

Easyprocedure

Find the angle whose adjacent is 6 cm and hypotenuse is 10 cm.

2 marks3 minsright-angled-trigonometry-q2

Show solution

Worked solution

1.Spot the skill: SOH-CAH-TOA: sin(θ) = O/H, cos(θ) = A/H, tan(θ) = O/A.
2.Use the write the trigonometric equation stage first, then apply the inverse sine.
3.Keep the final answer visible: 53.1°.

Final answer

53.1°

Mark points

M1: use the correct soh-cah-toa: sin(θ) = o/h, cos(θ) = a/h, tan(θ) = o/a. g. o = h × sin θ). g. θ = sin^-1(o/h)).label sides relative to the angle you are working with.
A1: 53.1°

Watch out

Mediumreasoning

A 5 m ramp rises 1.2 m. Find the angle it makes with the ground.

3 marks4 minsright-angled-trigonometry-q3

Show solution

Worked solution

1.Spot the skill: SOH-CAH-TOA: sin(θ) = O/H, cos(θ) = A/H, tan(θ) = O/A.
2.Use the apply the inverse sine stage first, then identify the known sides relative to the unknown angle.
3.Keep the final answer visible: ≈ 13.9°.

Final answer

≈ 13.9°

Mark points

M1: use the correct soh-cah-toa: sin(θ) = o/h, cos(θ) = a/h, tan(θ) = o/a. g. o = h × sin θ). g. θ = sin^-1(o/h)).label sides relative to the angle you are working with.
A1: ≈ 13.9°

Watch out

Hardproblem solving

Find the hypotenuse when opposite = 7 cm and angle = 42°.

3 marks5 minsright-angled-trigonometry-q4

Show solution

Worked solution

1.Spot the skill: SOH-CAH-TOA: sin(θ) = O/H, cos(θ) = A/H, tan(θ) = O/A.
2.Use the identify the known sides relative to the unknown angle stage first, then write the trigonometric equation.
3.Keep the final answer visible: ≈ 10.5 cm.

Final answer

≈ 10.5 cm

Mark points

M1: use the correct soh-cah-toa: sin(θ) = o/h, cos(θ) = a/h, tan(θ) = o/a. g. o = h × sin θ). g. θ = sin^-1(o/h)).label sides relative to the angle you are working with.
A1: ≈ 10.5 cm

Watch out

Exam-stylemulti-step

In a right triangle, adjacent = 9 cm and opposite = 12 cm. Find the angle and hypotenuse.

4 marks6 minsright-angled-trigonometry-q5

Show solution

Worked solution

1.Spot the skill: SOH-CAH-TOA: sin(θ) = O/H, cos(θ) = A/H, tan(θ) = O/A.
2.Use the write the trigonometric equation stage first, then apply the inverse sine.
3.Keep the final answer visible: θ ≈ 53.1°, hypotenuse = 15 cm.

Final answer

θ ≈ 53.1°, hypotenuse = 15 cm

Mark points

M1: use the correct soh-cah-toa: sin(θ) = o/h, cos(θ) = a/h, tan(θ) = o/a. g. o = h × sin θ). g. θ = sin^-1(o/h)).label sides relative to the angle you are working with.
A1: θ ≈ 53.1°, hypotenuse = 15 cm

Watch out

Grade 9 stretchproblem solving

A ladder is 6.5 m long and reaches 5.8 m up a wall. Find the angle between the ladder and the ground to 1 decimal place.

\theta

5.8\text{ m}

6.5\text{ m}

4 marks7 minsright-trig-g9

Show solution

Worked solution

1.The opposite side is 5.8 and the hypotenuse is 6.5.
2.Use sin( $\theta$ ) = opposite/hypotenuse.
3.Apply inverse sine.

Final answer

63.2 degrees

Mark points

M1: sin( $\theta$ ) = 5. $\frac{8}{6}$ .5
M1: inverse sine
A1: 63.2 degrees

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Right-angled trigonometry - 2 marksFind the side opposite a 35° angle in a right triangle with hypotenuse 20 cm.Mark answer

Answer

≈ 11.5 cm

2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer

Answer

3Properties of shapes - 2 marksName all 2D shapes with equal diagonals that bisect each other.Mark answer

Answer

Rectangle, square

4Perimeter, area and volume - 3 marksFind the perimeter of a rectangle with length 13 cm and width 8 cm.Mark answer

Answer

42 cm

Mastery check

I can explain the method for right-angled trigonometry.
I can show clear working without skipping key steps.
I can avoid this mistake: Students label the opposite and adjacent sides incorrectly by confusing which angle they are working from.Always label O, A and H relative to the angle marked with θ — not relative to any other angle in the triangle.

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