Find the value of cos two 𝐴 given cos of 𝐴 is negative three-fifths, where 𝐴 is between 90 degrees and 180 degrees, without using a calculator.
To find this value, we can use the formula cos two 𝐴 is equal to two cos squared 𝐴 minus one. And we can plug in the cosine of 𝐴 as negative three-fifths. So we plug in negative three-fifths for the cosine of 𝐴, and then we will have to square it. So if we square negative three, we get nine and if we square five, we get 25. So now we take two times nine twenty-fifths. So we take two times nine, and then we divide by 25. So two times nine is 18, and then we put it over 25 or divide by 25. And now we subtract one. When we subtract fractions, we need to have common denominators, so we can make one be twenty-five twenty-fifths because that is equal to one. So now we can subtract our numerators, and then keep our common denominator of 25. And we get negative seven twenty-fifths.
Now it also told us that 𝐴 is between 90 degrees and 180 degrees. So this will put us in quadrant number two out of the four quadrants because we’re between 90 degrees and 180 degrees. So we have 𝑥, 𝑦 labelled for a reason. Cos 𝜃 represents the 𝑥-value, so since we’re in the second quadrant, our 𝑥-value is negative so our cosine value should be negative. And indeed we did; we got a negative seven twenty-fifths.
So again, our final answer would be negative seven twenty-fifths.