In the given figure, find 𝜃. Give your answer to two decimal places.
In this question, we are given the lengths of all three sides of the triangle. And we are asked to calculate one of the angles labeled 𝜃. In order to calculate this, we will use the law of cosines, otherwise known as the cosine rule. This states that the cos of angle 𝐶 is equal to 𝑎 squared plus 𝑏 squared minus 𝑐 squared all divided by two 𝑎𝑏, where the lowercase letters represent the side lengths opposite the corresponding angles.
Substituting in the values from the figure, we have the cos of angle 𝜃 is equal to five squared plus seven squared minus 10 squared all divided by two multiplied by five multiplied by seven. The right-hand side simplifies to 25 plus 49 minus 100 divided by 70. This is equal to negative 13 over 35. We can then take the inverse cosine of both sides of this equation so that 𝜃 is equal to the inverse cos of negative 13 over 35.
Typing this into the calculator, we get 111.8037 and so on. We are asked to give our answer to two decimal places. Therefore, the three in the thousandths column is the deciding number. As this is less than five, we will round down. The angle 𝜃 to two decimal places is equal to 111.80 degrees.