Use the cosine rule in a non-right-angled triangle

The cosine rule

Use the cosine rule for two sides and their included angle, or use its rearranged form when all three sides are known. Calculate the squared quantity first and take a square root only when a side length is required

1. Use the included angle The angle in the side formula must be between the two known sides \(\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos A\)
2. Keep the cosine sign For an obtuse angle the cosine is negative, so subtracting the cosine term increases the squared side \(\displaystyle \cos120^{\circ}=-0.5,\qquad -2bc\cos120^{\circ}=+bc\)
3. Take the square root last The side formula first gives a squared side, so take the positive square root before reporting a length \(\displaystyle a^{2}=333\quad\Rightarrow\quad a=\sqrt{333}\approx18.25\)
4. Rearrange for an angle With three side lengths, subtract the opposite side square in the numerator and divide by twice the product of the other sides \(\displaystyle \cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\)

Worked example

A non-right-angled triangle has side b of 7 cm, side c of 10 cm and an included angle A of 70 degrees. Find side a to 2 decimal places

1. \(\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos A\)The two known sides and their included angle match the side form of the cosine rule
2. \(\displaystyle a^{2}=7^{2}+10^{2}-2(7)(10)\cos70^{\circ}\)Substitute the three given values with the subtraction in the formula
3. \(\displaystyle a^{2}\approx101.1172\ \mathrm{cm}^{2}\)Evaluate the right side while keeping enough digits for the square-root step
4. \(\displaystyle a=\sqrt{101.1172}\approx10.06\ \mathrm{cm}\)Take the positive square root and round the side length to 2 decimal places

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