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SQA National 5 Mathematics

SOH CAH TOA

Using sine, cosine and tangent in right-angled triangles.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Mark the right angle.
  • Label the side opposite the angle.
  • Choose sine, cosine or tangent from the sides involved.
National 5 Mathematics lesson

Explanation

SOH CAH TOA is used in right-angled triangles when an angle and sides are involved.

Sine uses opposite and hypotenuse, cosine uses adjacent and hypotenuse, and tangent uses opposite and adjacent.

If finding an angle, use the inverse trig button on a calculator.

Visual support

SOH CAH TOA

adjacentoppositehypotenuseθ

Key formulae and rules

  • sin θ = opposite / hypotenuse
  • cos θ = adjacent / hypotenuse
  • tan θ = opposite / adjacent

Watch out

Using SOH CAH TOA on a triangle that is not right-angled.

Check

Check your answer against the diagram. The longest side should still be opposite the largest angle.

Exam tip

Write the trig ratio before rearranging. This earns method marks and reduces calculator mistakes.

Calculator tip

Make sure the calculator is in degree mode before using sin, cos, tan or inverse trig.

Worked examples

Worked example 1

Find a missing side

A right-angled triangle has angle 35 degrees and hypotenuse 12 cm. Find the opposite side.

  1. Use sin because opposite and hypotenuse are involved.
  2. sin 35 = x / 12
  3. x = 12 sin 35
  4. x = 6.88

Answer: The opposite side is 6.9 cm to 1 decimal place.

Worked example 2

Find a missing angle

The opposite side is 7 cm and the adjacent side is 10 cm. Find the angle.

  1. Use tan because opposite and adjacent are involved.
  2. tan θ = 7 / 10
  3. θ = tan−1(0.7)
  4. θ = 34.99..

So: The angle is 35.0 degrees to 1 decimal place.

Worked example 3

Find the adjacent side

A right-angled triangle has angle 42 degrees and hypotenuse 15 m. Find the adjacent side.

  1. Use cos because adjacent and hypotenuse are involved.
  2. cos 42 = x / 15
  3. x = 15 cos 42

Answer: x = 11.1 m to 1 decimal place

Watch out

  • Using SOH CAH TOA on a triangle that is not right-angled.
  • Choosing the side adjacent to the right angle instead of adjacent to the marked angle.
  • Forgetting inverse trig when finding an angle.