And the Cosine Rule: Its easier than it seems

The cosine rule has 4 variables or unknowns:

side a, side b, side c and an angle.

Cosine Rule

c2=a2+b22abcosC

Or, to find an angle:

cosC=a2+b2c22ab

This means that if you are given 3 of those unknowns, you need to use the cosine rule.

E.g. if you are given 3 sides of a triangle (remember it should not be a right angled triangle.)

Use the cosine rule to find the angle.

Or if you are given 2 sides and the angle IN BETWEEN THOSE 2 SIDES then use the cosine rule to find the 3rd side.

Question 1: Finding a Missing Side

A triangle has sides of length 8 cm and 11 cm, with an included angle of 120°. Calculate the length of the third side to 3 significant figures.

Solution:

We use the cosine rule:

c2=a2+b22abcosC

Substituting values:

c2=82+1122(8)(11)cos120

c2=64+1212(8)(11)(0.5)

c2=64+121+88=273

c=273=16.5 cm (to 3 significant figures)

Final Answer: 16.5 cm


Question 2: Finding an Angle

A triangle has sides of length 10 cm, 12 cm, and 15 cm. Find the largest angle in the triangle to 1 decimal place.

Solution:

The largest angle is opposite the longest side, so we find angle A using the cosine rule:

cosA=b2+c2a22bc

Substituting values:

cosA=102+1221522(10)(12)

cosA=100+144225240

cosA=19240

A=cos1(19240)

A=85.5

Final Answer: 85.5°


Question 3: Word Problem – Bearings

Two boats, A and B, leave a port at the same time.

  • Boat A travels 20 km due east.
  • Boat B travels 25 km on a bearing of 40° from the port.

Find the distance between the two boats to 1 decimal place.

Solution:

We form a triangle where:

  • a=20 km
  • b=25 km
  • Included angle C=40

Using the cosine rule:

c2=a2+b22abcosC

Substituting values:

c2=202+2522(20)(25)cos40

c2=400+6252(20)(25)(0.766)

c2=1025766=259

c=259=16.1 km

Final Answer: 16.1 km

Interactive Cosine Rule Triangle Calculator - beta mode

Cosine Rule Triangle Calculator - beta mode

Enter two sides and the included angle (in degrees):

More examples and practice questions coming soon!

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The Sine Rule: when to use it and how to do the hard questions