The cosine rule

The Maths Habit

Two sides of a triangle are 6 cm and 8 cm. How long could the third side be?

Anything between 2 cm and 14 cm: it depends on the angle between them

Sides 6 cm and 8 cm with 90° between them. Find x.

x2 = 62 + 82 = 100, so x = 10 cm

Keep the sides 6 cm and 8 cm. Close the angle, then open it. What happens to x?

60°: x = 7.2 cm, shorter than 10 cm

120°: x = 12.2 cm, longer than 10 cm

x2 = 62 + 82 − 2 × 6 × 8 × cos A

Angle Acos A2 × 6 × 8 × cos Ax2 = 100 − that
60°0.54852
90°00100
120°−0.5−48148

At 90°, cos A = 0 and it is Pythagoras. The cosine rule corrects Pythagoras for the angle.

a2 = b2 + c2 − 2bc cos A

Which rule?

Two sides and the angle between them Cosine rule

Which rule?

Two angles and one side Sine rule

Sides 6 cm and 8 cm with 50° between them. Find the third side, x.

x2 =62+82
− 2 ×6×8× cos50°
x2 = 38.29
Answer6.2 cm

Sides 7 cm and 4 cm with 65° between them. Find the third side, x.

x2 =72+42
− 2 ×7×4× cos65°
x2 = 41.33
Answer6.4 cm

Sides 6 cm and 8 cm with 50° between them. Find the third side, x.

x2 =62+82
− 2 ×6×8× cos50°
x2 = 38.29
Answer6.2 cm

Sides 5 cm and 9 cm with 120° between them. Find the third side, x.

x2 =52+92
− 2 ×5×9× cos120°
x2 = 151
Answer12.3 cm

The cosine rule worksheet, page 1

Answer sheet

  1. Two sides of a triangle are 9 cm and 12 cm with 52° between them. Find the perimeter.

    30.6 cm

  2. A triangle has sides 5 cm, 5 cm and 8 cm. Is its largest angle acute or obtuse?

    Obtuse: 106.3°

  3. A triangle has sides 6 cm, 9 cm and 11 cm. Find its smallest angle to 1 decimal place.

    33.0°

  1. Find the length x.

    x = 7 cm
    8 cm5 cmx60°

  2. Find angle x to 1 decimal place.

    x = 86.4°
    6 cm4 cm7 cmx

  3. A triangle has sides of 5 cm, 7 cm and 10 cm. Find its largest angle to 1 decimal place.

    111.8°
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