Cosine Rule

The Cosine Rule: Finding an Unknown Side

Two sides of the triangle and the included angle (i.e., the angle between the two sides).

The Cosine Rule for finding an unknown side is:

a^2=b^2+c^2−2bc⋅cos⁡A

Where:

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Example: Find the length of side $x$ in the triangle where $b=:highlight[5.2 m]$ , $c=:highlight[4.5m]$ , and $∠A=:highlight[58\degree]$ .

x^2=5.2^2+4.52−2×5.2×4.5×cos⁡58\degree

x^2=27.04+20.25−23.4×0.5299

x^2=22.48977 m^2

x=\sqrt{22.48977}≈:success[4.74 \ m (2dp)]

So, the length of side $x$ is approximately 4.74\ m.

The Cosine Rule for finding an unknown angle is:

cos\ ⁡A=\frac{b^2+c^2−a^2}{2bc}

Where:

infoNote

Example: Find the size of angle $x$ in the triangle where $a=:highlight[9 m]$ , $b=:highlight[11 m]$ , and $c=:highlight[12 m]$ .

cos⁡\ x=\frac{11^2+12^2−9^2}{2×11×12}

cos\ ⁡x=\frac{121+144−81}{264}=\frac{184}{264}≈0.69697

x=cos^{⁡−1}(0.69697)≈:success[72.97\degree (2dp)]

So, the angle is approximately 72.97°