Mini-Lesson

Trigonometry

This mini-lesson covers Trigonometry for Cambridge IGCSE Mathematics (0580): Pythagoras, SOHCAHTOA, the sine rule, cosine rule, area = ½ab sin C, 3D trigonometry and angles of elevation & depression (Extended content included).

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Pythagoras

Pythagoras' theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

a² + b² = c²c is the hypotenuse — the longest side, opposite the right angle

Worked example

Legs 6 and 8: c = √(6² + 8²) = √100 = 10.

Finding a shorter side? Subtract: a = √(c² − b²).

Calculate

Your turn

1A right-angled triangle has the two shorter sides 5 cm and 12 cm. Work out the length of the hypotenuse.

cm

Hint: √(5² + 12²) = √(25 + 144) = √169.

SOHCAHTOA

Trigonometric ratios

For right-angled triangles, relate an angle to the sides:

sin = O/H · cos = A/H · tan = O/AO = opposite, A = adjacent, H = hypotenuse

Worked example

Opposite 3, hypotenuse 6: sin θ = 3/6 = 0.5, so θ = sin⁻¹(0.5) = 30°.

Exact values: sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1 (Extended).

Calculate

Your turn

2In a right-angled triangle the side opposite a 30° angle is 5 cm. The hypotenuse H satisfies sin 30° = 5/H, and sin 30° = 0.5. Work out H.

cm

Hint: 0.5 = 5/H, so H = 5 ÷ 0.5.

Quick check

?Which ratio would you use to find the opposite side when you know the adjacent side and the angle?

Sine & cosine rules

For any triangle (not just right-angled):

a/sin A = b/sin B = c/sin Ccosine rule: a² = b² + c² − 2bc·cos A

Use the sine rule when you have a side and its opposite angle. Use the cosine rule for two sides + the included angle, or all three sides.

Area of any triangle: Area = ½ab sin C (two sides and the included angle).

Calculate

Your turn

3A triangle has sides a = 5 and b = 8 with an included angle C = 30°. Using Area = ½ab sin C and sin 30° = 0.5, work out the area.

cm²

Hint: ½ × 5 × 8 × 0.5 = 20 × 0.5.

Calculate

Your turn

4In triangle ABC, b = 7, c = 24 and angle A = 90°. Using the cosine rule a² = b² + c² − 2bc·cos A with cos 90° = 0, work out side a.

cm

Hint: cos 90° = 0, so a² = 7² + 24² = 49 + 576 = 625.

Elevation, depression & 3D

The angle of elevation is measured up from the horizontal; the angle of depression is measured down. They are equal for the same line of sight (alternate angles).

For 3D problems, find a right-angled triangle inside the solid — often using a diagonal — then apply Pythagoras or SOHCAHTOA.

Worked example

Base diagonal of a 3-by-4 rectangle: √(3² + 4²) = 5, then use that as one side of the vertical triangle.

Calculate

Your turn

5A cuboid has a base 3 cm by 4 cm. Work out the length of the diagonal of the base.

cm

Hint: base diagonal = √(3² + 4²) = √25.

Quick check

?The angle of elevation of a tower top from a point is 40°. What is the angle of depression from the tower top back down to that point?

Match game

Ratio ⇄ definition

Tap a trig ratio, then its SOHCAHTOA definition.

Sort it

Right-angled triangle only?

Sort each method: does it need a right angle, or does it work for any triangle? Tap it, then its box.

∟ Needs right angle

△ Any triangle

Recap

Key points

Pythagoras: a² + b² = c² (right-angled triangles).

SOHCAHTOA: sin = O/H, cos = A/H, tan = O/A.

Sine rule: a/sin A = b/sin B = c/sin C.

Cosine rule: a² = b² + c² − 2bc·cos A.

Area: ½ab sin C. Elevation = depression (same line of sight).

You've covered the Trigonometry essentials for Cambridge IGCSE Maths (0580). Press Finish to see your score.

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