Expert: Sherry Wallin - 8/30/2009

Question
Suppose a = cos(20). Find the following in terms of 'a':

1) sin(20)
2) cos(70)

I'm having trouble on how to start this? What are the steps that I need to take to solve this? Thanks for the help.

Answer
Sue Lee~
    The sine of an angle and the cosine of an angle are related. I like to think of them as cousins. If the sum of their angles adds up to 90 degrees then they are complementary to each other and they have the same value! As an example, suppose the sin 45 is [sqrt2]/2, then the cos 45 is [sqrt 2]/2 since 45 + 45 = 90. I chose this simple example because you probably already know that when the angle is 45 degrees that is the only time the sin theta = cos theta. But if the sin 30 = 1/2 then the
cos 60 = 1/2 because 30 + 60 = 90. From all that I have said above you should be able to see that the sin 20 = cos 70 since 20 + 70 = 90. All that remains is for you to figure out how either of those are related to a = cos 20. You know that cos^2 theta + sin^2 theta is always 1, correct?
So cos^2(20) + sin^2 (20) = 1 and since a = cos(20) we have a^2 + sin^2 (20) = 1, right? Which means sin^2(20) = 1 - a^2 and thus the sin 20 = +-sqrt(1-a^2) and since sin 20 = cos 70, the cos 70 is also equal to +-sqrt(1-a^2).

Math Prof