Cosine rule

By Martin McBride, 2023-03-17
Tags: triangle cosine cosine rule sine rule solving triangles
Categories: gcse trigonometry

The cosine rule is a trigonometry formula that relates the sides and angles of a triangle. It can be used to solve a triangle if we know either:

  • Two sides of the triangle, and the angle enclosed between those sides.
  • Three sides but none of the angles.

For other cases, you will need to use the sine rule.

The rule applies to any triangle, not just right-angled triangles.

Labelling the triangle

It is important to label the triangle correctly, otherwise the rule won't work! We name the angles A, B and C, and we name the sides a, b and c:

Labelling a triangle

The important thing to remember is that each angle is opposite the side of the same name:

  • Angle A is opposite side a.
  • Angle B is opposite side b.
  • Angle C is opposite side c.

The cosine rule

The cosine rule tells us that:

Labelling a triangle

Finding a side using the cosine rule

In this example, we know the sides b and c, plus the enclosed angle A, and we wish to find the other side, a.

Labelling a triangle

The cosine rule lets us find a squared, so to find a we need to take the square root of both sides. Here is the formula:

Finding a side using cosine rule

Finding an angle using the cosine rule

In this example we know the sides a, b and c, so we can find the angle, A:

Labelling a triangle

The cosine rule can be rearranged to find the cosine of A. Here is the formula:

Finding an angle using cosine rule

In both cases, we will know the value of one angle and all three sides. We can then use the sine rule to find the other angles.

See also

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