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Maths

Further trigonometry - Higher

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The cosine rule

We can use the cosine formula to find the length of a side or size of an angle.

For a triangle with sides a,b and c and angles A, B and C the cosine rule can be written as:

  • a2 = b2 + c2 - 2bc cos A
  • or
  • b2 = a2 + c2 - 2ac cos B
  • or
  • c2 = a2 + b2 - 2ab cos C

These formulae can be rearranged to give :

cos A = b to the power 2 + c to the power 2 - a to the power 2 over 2 b c

cos B = a to the power 2 + c to the power 2 - b to the power 2 divided by 2 a c

cos C = a to the power 2 + b to the power 2 - c to the power 2

When to use it

We can use the cosine formula when we are given:

Two sides and an angle.

image: triangle

Three sides

Check whether your formula sheet gives these formulae in both formats. If it does not, you may need to rearrange them yourself.

Question

Find the length of BC.

toggle answer

Answer

Did you get 4.86cm? If so, well done.

If not, remember to use the formula:

a2 = b2 + c2 - 2bc cos A

Substitute the values into the formula.

a2 = 72 + 32 - 2 × 3 × 7 cos 35

a2 = 58 - 42 cos 35

a2 = 23.5956

a = 4.86cm

Question

Find the size of angle R.

toggle answer

Answer

R = 31.8°

You need to use the formula: cos R = p to the power 2 + q to the power 2 - r to the power 2

OS = 10 over tan 35 degrees

R = inv cos 0.8497

Therefore, R = 31.8°

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Back to Geometry and measures index

Activity

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Angles

Now to take a different angle on revision with this activity!

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