Trigonometry: Solving Triangles & Angles of Elevation

Using sin, cos and tan to find a missing side or angle in a right-angled triangle, and applying them to angles of elevation and depression in real problems.

Once you know SOH-CAH-TOA, every right-angled triangle problem follows the same three steps: label the sides, choose the ratio, solve.

1Choosing the right ratio

sin θ = opp / hyp · cos θ = adj / hyp · tan θ = opp / adj

Label the sides relative to the angle you are given or want: opposite, adjacent, hypotenuse.
Tick the two sides involved in the question — the one you know and the one you want.
The ratio containing both of those sides is the one to use.

Worked ExampleFind x: the side opposite a 40° angle, with hypotenuse 15 cm

1
Opposite and hypotenuse are involved, so use sin.
SOH.
2
sin 40° = x ÷ 15.
3
x = 15 × sin 40° = 9,64 cm.
Multiply across; round as asked.

Worked ExampleFind the angle: opposite = 7, adjacent = 10

1
Opposite and adjacent → tan.
TOA.
2
tan θ = 7 ÷ 10 = 0,7.
3
θ = tan⁻¹(0,7) = 34,99° ≈ 35°.
Use the inverse function to get back to the angle.

⚠️

To find an angle you need the inverse (sin⁻¹, cos⁻¹, tan⁻¹). Typing tan(0,7) instead of tan⁻¹(0,7) gives a completely wrong answer.

2Angles of elevation and depression

An angle of elevation of 34° from an observer 120 m from a tower. tan 34° = h ÷ 120.

The angle of elevation is measured upwards from the horizontal.
The angle of depression is measured downwards from the horizontal.
They are equal when measured between the same two points (alternate angles on parallel horizontals).

Worked ExampleFrom 120 m away the angle of elevation to a tower top is 34°. Find its height.

1
Sketch it: the height is opposite the angle, 120 m is adjacent.
Always draw the triangle first.
2
Opposite and adjacent → tan: tan 34° = h ÷ 120.
TOA.
3
h = 120 × tan 34° = 80,94 m.

💡

Draw the triangle before touching the calculator. Almost every mark lost in this topic comes from choosing the wrong ratio, not from arithmetic.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Find x: opposite a 40° angle, hypotenuse 15 cm.
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15 sin 40° = 9,64 cm
2
Find x: adjacent to a 55° angle, hypotenuse 20 cm.
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20 cos 55° = 11,47 cm
3
Find x: opposite a 30° angle, adjacent 12 cm.
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12 tan 30° = 6,93 cm
4
Find θ: opposite 7, adjacent 10.
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tan⁻¹(0,7) = 35,0°
5
Find θ: opposite 5, hypotenuse 13.
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sin⁻¹(5/13) = 22,6°
6
Find θ: adjacent 8, hypotenuse 17.
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cos⁻¹(8/17) = 61,9°
7
From 120 m away the elevation to a tower top is 34°. Find its height.
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120 tan 34° = 80,94 m
8
What is the difference between an angle of elevation and one of depression?
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Elevation is measured up from the horizontal; depression is measured down.