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:
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:
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.
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 an angle using the cosine rule
In this example we know the sides a, b and c, so we can find the angle, A:
The cosine rule can be rearranged to find the cosine of A. Here is the formula:
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.
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