Find the measure of angle 𝐴 to one decimal place.
In this question, we’re given the lengths of all three sides of our triangle. And we need to calculate the measure of one of the angles. We can do this using the law of cosines. This states that 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 multiplied by the cos of angle 𝐴. The lowercase letters 𝑎, 𝑏, and 𝑐 are the side lengths opposite the corresponding angles.
This formula can be rearranged to help us calculate the measure of angle 𝐴. Firstly, we subtract 𝑎 squared from both sides. And we can also add two 𝑏𝑐 multiplied by the cos of angle 𝐴 to both sides. This gives us the equation two 𝑏𝑐 multiplied by the cos of angle 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared. Dividing both sides of this equation by two 𝑏𝑐 gives us the cos of angle 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared all divided by two 𝑏𝑐. It is worth recalling this version of the law of cosines when we need to calculate an angle and are given the three side lengths of a triangle.
Our next step is to substitute the values of lowercase 𝑎, 𝑏, and 𝑐 into this equation. The cos of angle 𝐴 is equal to 11 squared plus 20 squared minus 20 squared all divided by two multiplied by 11 multiplied by 20. 20 squared minus 20 squared is equal to zero. This leaves us with 11 squared on the numerator and two multiplied by 11 multiplied by 20 on the denominator. Next, we can divide the numerator and denominator by 11. The cos of angle 𝐴 is therefore equal to 11 over 40.
Our next step is to take the inverse cosine of both sides of our equation. This gives us that angle 𝐴 is equal to the inverse cos of 11 over 40. Typing the right-hand side into our calculator gives us 74.0379 and so on. We are asked to give our answer to one decimal place. And as the second digit after the decimal point is less than five, we will round down. The measure of angle 𝐴 correct to one decimal place is 74.0 degrees.