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To find the missing angle of a triangle using the cosine rule, you need to know the length of all three sides of the triangle. For example, you might have triangle RST. Side SR is 8 cm long. Side ST is 10 cm long. Side RT is 12 cm long. What is the measurement of angle S? The formula is c2=a2+b2−2abcos⁡C{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos {C}}. In this formula, cos⁡C{\displaystyle \cos {C}} equals the cosine of the angle you are trying to find. The variable c{\displaystyle c} equals the side opposite the missing angle. The variables a{\displaystyle a} and b{\displaystyle b} are the lengths of the other two sides. Plug these values into the formula. For example, since side RT is opposite the missing angle, angle S, side RT will equal c{\displaystyle c} in the formula. The other two side lengths will be a{\displaystyle a} and b{\displaystyle b}. It doesn’t matter which side is which variable. So, your formula should look like this: 122=82+102−2(8)(10)cos⁡C{\displaystyle 12^{2}=8^{2}+10^{2}-2(8)(10)\cos {C}}. You are multiplying 2ab{\displaystyle 2ab} times the cosine of the missing angle, which you don’t know yet. So, the variable should remain. For example, 122=82+102−160cos⁡C{\displaystyle 12^{2}=8^{2}+10^{2}-160\cos {C}}. Remember that to square a number, you multiply the number by itself. For example, 144=82+102−160cos⁡C{\displaystyle 144=8^{2}+10^{2}-160\cos {C}} Make sure you square each number first, and then add them together. For example:144=64+100−160cos⁡C{\displaystyle 144=64+100-160\cos {C}}144=164−160cos⁡C{\displaystyle 144=164-160\cos {C}} To do this, subtract the sum of a2{\displaystyle a^{2}} and b2{\displaystyle b^{2}} from both sides of the equation. Then, divide each side of the equation by the coefficient of the missing angle’s cosine. For example, to isolate the cosine of the missing angle, subtract 164 from both sides of the equation, then divide each side by -160:144−164=164−164−160cos⁡C{\displaystyle 144-164=164-164-160\cos {C}}−20=−160cos⁡C{\displaystyle -20=-160\cos {C}}−20−160=−160cos⁡C−160{\displaystyle {\frac {-20}{-160}}={\frac {-160\cos {C}}{-160}}}0.125=cos⁡C{\displaystyle 0.125=\cos {C}} This will give you the measurement of the missing angle. On a calculator, the inverse cosine key is denoted by COS−1{\displaystyle COS^{-1}}. For example, the inverse cosine of .0125 is 82.8192. So, the missing angle, angle S, is 82.8192 degrees.
Assess what values you know. Set up the formula for the Cosine Rule. Determine the values of a{\displaystyle a}, b{\displaystyle b}, and c{\displaystyle c}. Complete the necessary multiplication. Find the square of c{\displaystyle c}. Add the squares of a{\displaystyle a} and b{\displaystyle b}. Isolate the cosine of the missing angle. Find the inverse cosine.