Step-by-step worked examples and graded practice questions on trigonometry — SOHCAHTOA, finding sides and angles, angles of elevation and depression, and exact values.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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In a right-angled triangle, the three sides are named relative to a chosen angle (not the right angle): the hypotenuse is always the longest side, the opposite side is across from the angle, and the adjacent side is next to the angle (but isn't the hypotenuse).
The three formula triangles below are a standard way to remember SOHCAHTOA — cover up the letter you want to find, and the other two show you the calculation (side by side means divide, e.g. covering S leaves O over H).
To find a side, cover its letter: e.g. for Sin, covering H leaves O next to S — read as O ÷ H. To rearrange for the top letter, the two bottom letters multiply instead: O = S × H.
Worked Example 1 — Finding a side
A right-angled triangle has an angle of 30° and a hypotenuse of 10cm. Find the length of the side opposite the angle.
1
The opposite and hypotenuse are involved, so use Sin = Opposite ÷ Hypotenuse.
The angle is always marked with an arc — the hypotenuse is opposite the right angle, and "opposite"/"adjacent" are defined relative to the marked angle.
Finding a missing angle
If two sides are known, use the inverse trig function (sin⁻¹, cos⁻¹ or tan⁻¹) to find the angle.
Worked Example 2 — Finding an angle
A right-angled triangle has a hypotenuse of 10cm and an opposite side of 6cm. Find the angle θ.
1
The opposite and hypotenuse are involved, so use Sin = Opposite ÷ Hypotenuse.
2
sin(θ) = 6 ÷ 10 = 0.6
3
θ = sin⁻¹(0.6) = 36.9° (1 d.p.)
Answer36.9°
Here θ (the angle) is unknown — the two known sides tell you which trig ratio to use.
Applications: elevation, depression and non-right-angled triangles
Trigonometry also solves real-world problems involving angles of elevation and depression (measured from the horizontal), and — using different formulae — problems involving triangles with no right angle at all.
Worked Example 3 — Angle of elevation
From a point 50m from the base of a tower, the angle of elevation to the top of the tower is 32°. Find the height of the tower.
1
Sketch the triangle: the height is opposite the angle, the 50m distance is adjacent to it.
2
tan(32°) = height ÷ 50, so height = 50 × tan(32°)
3
height = 50 × 0.625 = 31.2m (1 d.p.)
Answer31.2m
The angle of elevation is measured upward from the horizontal ground.
Worked Example 4 — Non-right-angled triangle area
Find the area of a triangle with two sides of 6cm and 9cm, and an included angle of 40°.
1
Area = ½ab sin(C), where C is the angle between the two given sides.
2
Area = ½ × 6 × 9 × sin(40°) = 27 × 0.643
3
Area = 17.4cm² (1 d.p.)
Answer17.4cm²
Exact trigonometric values
At Higher tier and Grade 8–9, you're expected to know these values without a calculator.
Angle
0°
30°
45°
60°
90°
sin
0
½
√2⁄2
√3⁄2
1
cos
1
√3⁄2
√2⁄2
½
0
tan
0
√3⁄3
1
√3
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Practice questions
Work through each question before checking the answers. Where a question describes a real-world or non-right-angled scenario, the diagram shown is required to answer it.
Foundation (Grade 3–5)
Q1A right-angled triangle has an angle of 35° and a hypotenuse of 12cm. Find the opposite side (1 d.p.).Foundation
Diagram for Question 1 (not to scale).
Q2A right-angled triangle has an angle of 50° and a hypotenuse of 9cm. Find the adjacent side (1 d.p.).Foundation
Q3A right-angled triangle has an angle of 40° and an adjacent side of 7cm. Find the opposite side (1 d.p.).Foundation
Q4A right-angled triangle has an opposite side of 5cm and a hypotenuse of 10cm. Find the angle θ.Foundation
Diagram for Question 4 (not to scale).
Q5A right-angled triangle has an opposite side of 6cm and an adjacent side of 8cm. Find the angle θ.Foundation
Higher (Grade 5–7)
Q6A right-angled triangle has an angle of 28° and an opposite side of 6cm. Find the hypotenuse (1 d.p.).Higher
Q7From a point 60m from the base of a lighthouse, the angle of elevation to the top is 40°. Find the height of the lighthouse (1 d.p.).Higher
Q8A right-angled triangle has an adjacent side of 9cm and a hypotenuse of 15cm. Find the angle θ.Higher
Q9A right-angled triangle has an opposite side of 5cm and an adjacent side of 12cm. Find the angle θ.Higher
Q10The diagram shows a cliff and a boat. Find the horizontal distance from the base of the cliff to the boat (1 d.p.).Higher
Diagram for Question 10 (not to scale). The dashed line shows the horizontal from which the angle of depression is measured.
Higher — Hard (Grade 8–9)
Q11Without a calculator, find the exact value of sin(60°) × cos(30°).Grade 8–9
Q12A ladder makes an angle of 65° with the ground and reaches 8m up a wall. Find the length of the ladder (1 d.p.).Grade 8–9
Q13The diagram shows a hill with a vertical pole on top, viewed from a point 60m from the base of the hill. Find the height of the pole (1 d.p.).Grade 8–9
Diagram for Question 13 (not to scale). The pole (teal) sits on top of the hill (grey). Both angles are measured from the horizontal ground.
Q14Solve tan(θ) = 1 for 0° ≤ θ ≤ 90°, without a calculator.Grade 8–9
Q15The diagram shows a triangle with two sides of 8cm and 10cm, and an included angle of 50°. Find its area (1 d.p.).Grade 8–9
Diagram for Question 15 (not to scale).
Answers
Foundation (Q1–Q5)
Q16.9cm(12 × sin35°)
Q25.8cm(9 × cos50°)
Q35.9cm(7 × tan40°)
Q430°(sin⁻¹(5/10))
Q536.9°(tan⁻¹(6/8))
Higher (Q6–Q10)
Q612.8cm(6 ÷ sin28°)
Q750.4m(60 × tan40°)
Q853.1°(cos⁻¹(9/15))
Q922.6°(tan⁻¹(5/12))
Q1085.8m(40 ÷ tan25°)
Higher — Hard (Q11–Q15)
Q11¾(√3⁄2 × √3⁄2 = 3⁄4)
Q128.8m(8 ÷ sin65°)
Q1314.0m(60tan35° − 60tan25° = 42.0 − 28.0)
Q14θ = 45°
Q1530.6cm²(½ × 8 × 10 × sin50°)
Common mistakes
Common Mistake 1
Labelling opposite and adjacent from the wrong angle
Opposite and adjacent are only meaningful relative to the marked angle — if the angle moves to the other non-right vertex, the same two sides swap roles.
Common Mistake 2
Forgetting to use the inverse function when finding an angle
If sin(θ) = 0.6, the angle is θ = sin⁻¹(0.6), not sin(0.6). Mixing these up is a very common calculator error.
Common Mistake 3
Measuring elevation and depression from the wrong line
Both angles are measured from a horizontal line, never from the vertical or from the line of sight itself.
Common Mistake 4
Using SOHCAHTOA on a triangle with no right angle
SOHCAHTOA only works in right-angled triangles. For any other triangle, use the Sine Rule, Cosine Rule, or the ½ab sin(C) area formula instead.
Exam tips
💡 Exam Tip 1
Label the triangle before choosing a ratio
Mark which side is the hypotenuse, opposite and adjacent relative to the given angle first — this makes choosing SOH, CAH or TOA automatic.
💡 Exam Tip 2
Check your calculator is in degree mode
A calculator left in radian mode gives completely wrong answers for GCSE trigonometry — check for "DEG" in the display before starting.
💡 Exam Tip 3
Learn the exact values rather than memorising decimals
Grade 8–9 questions expect exact answers like √3⁄2, not calculator decimals — practising the 0°/30°/45°/60°/90° table pays off.
💡 Exam Tip 4
Sketch multi-stage problems as two triangles
For problems like a pole on a hill, draw both right-angled triangles sharing the same base, and subtract the two heights at the end.
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