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Geometry & Measures · Foundation & Higher

Trigonometry (SOH-CAH-TOA)

Trigonometry (trig) is used to find missing sides and angles in right-angled triangles using the sine, cosine and tangent ratios.

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Key facts to remember

  • 1SOH: sin θ = Opposite ÷ Hypotenuse
  • 2CAH: cos θ = Adjacent ÷ Hypotenuse
  • 3TOA: tan θ = Opposite ÷ Adjacent
  • 4Opposite = side opposite the angle; Adjacent = side next to the angle (not hyp); Hypotenuse = longest side.
  • 5To find an angle: use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹).
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Formulas

Sine
sin θ = O/H
Cosine
cos θ = A/H
Tangent
tan θ = O/A
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Worked examples

Example 1

Find the side x opposite a 35° angle in a right-angled triangle with hypotenuse 12 cm.

Working

  1. We have angle (35°), hypotenuse (12), and want opposite → use SIN
  2. sin 35° = x/12
  3. x = 12 × sin 35°
  4. x = 12 × 0.5736 = 6.88 cm (3 s.f.)
Answerx ≈ 6.88 cm
Example 2

Find angle θ in a right-angled triangle where opposite = 7 cm and adjacent = 9 cm.

Working

  1. We have opposite and adjacent → use TAN
  2. tan θ = 7/9
  3. θ = tan⁻¹(7/9)
  4. θ = 37.9° (3 s.f.)
Answerθ ≈ 37.9°
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Common mistakes

✗Confusing opposite and adjacent — always label the triangle relative to the given angle.
✗Using sin when you should use cos (mixing up which sides are involved).
✗Forgetting to use the inverse function when finding an angle.
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Exam tips

✓Always label O, A and H on your diagram before choosing sin, cos or tan.
✓Make sure your calculator is in DEGREE mode, not RADIAN mode.

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