Find the value of cos 𝜃 over two given cos 𝜃 is equal to 15 over 17, where 𝜃 is between zero degrees and 90 degrees without using a calculator.
The first thing we’re going to write down is the half-angle identity: cos 𝜃 over two is equal to plus or minus the square root of one plus cos 𝜃 over two. This holds for all values of 𝜃.
Substituting the value of cos 𝜃 that we’re given in the question, 15 over 17, we get that cos 𝜃 over two is equal to plus or minus the square root of one plus 15 over 17 over two. This plus or minus sign tells us that we narrowed down the value of cos 𝜃 over two to two possibilities, but we need to do a bit more work to decide which of these values cos 𝜃 over two actually is.
We are given another piece of information, which will help us make our decision. We’re told that 𝜃 lies between zero degrees and 90 degrees. For this range of 𝜃, we can see that cos 𝜃 is always positive. As 𝜃 lies between zero and 90 degrees, 𝜃 over two must lie between zero and 45 degrees. And we see that in this range, cos 𝜃 over two must also be positive.
This means that we can confidently pick the positive value of cos 𝜃 over two. Now it’s just a case of simplifying this. There are many ways to do this. I’ve chosen to multiply by 17 over 17, which is otherwise known as one inside the radical. That gets me the square root of 17 plus 15 over 34.
When I find the numerator is 32, I see that there’s a common factor of two. So I divide out by that to get the square root of 16 over 17, which is four over root 17. Cos 𝜃 of four over two is four over root 17. We multiply by the square root of 17 over the square root of 17 to rationalise the denominator . And that gives our final answer cos 𝜃 over two is equal to four root 17 over 17.