Choose the rule a triangle problem needs

Choosing a triangle rule

Choose from the given data, not from the name of the unknown quantity. Check the side-angle relationships, whether a right angle is present and whether the question asks for an area

1. Use the sine rule Use a known side-opposite-angle pair and another required part when solving a side or angle \(\displaystyle \frac{a}{\sin A}=\frac{c}{\sin C}\)
2. Use the cosine rule Use two sides and their included angle, or all three side lengths, when solving a side or angle \(\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos A\)
3. Use a right-triangle rule When a right angle is present, use a trigonometric ratio for an angle and side, or Pythagoras for side lengths \(\displaystyle \sin\theta=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}},\qquad a^{2}+b^{2}=c^{2}\)
4. Choose the area rule from the data Use a base and perpendicular height, two sides and their included angle, or all three sides for the matching area rule \(\displaystyle K=\frac{1}{2}bh,\quad K=\frac{1}{2}ab\sin C,\quad K=\sqrt{s(s-a)(s-b)(s-c)}\)

Worked example

A non-right-angled triangle has the measurements shown. Which rule should be used to find the third side?

1. \(\displaystyle \text{given: two sides and their included angle}\)Read the available measurements before considering the unknown side
2. \(\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos A\)This is the side form of the cosine rule for two sides and their included angle
3. \(\displaystyle \boxed{\text{cosine rule}}\)Select the cosine rule because all of its required inputs are given

Common mistakes

Difficulty levels

Questions about this topic

In this chapter