So we are talking about trigonometric identities and one of the most important is the Pythagorean Identity; Cosine squared theta plus sine squared theta equals 1. And this identity comes in two other forms, sine squared theta equals 1 minus cosine square theta and cosine square theta equals I minus sine squared theta.
I’ll get a chance to use one these in the next example here’s the problem. If sine of theta equals 3/5 and theta is between pi over 2 and pi, find cosine theta and tangent theta. So first of all let me find cosine theta using a Pythagorean identity; Cosine squared theta equals 1 minus sine squared theta. Sine of theta is 3/5 so I have 1 minus 3/5 squared. And 3/5 squared is 9 over 25, 1 minus 9 over 25, 1 minus 9 over 25 is 25 over 25 minus 9 over 25 and that’s 16 over 25.
So cosine squared theta equals 16 over 25. That means cosine theta is plus or minus 4/5. In order to determine whether there is plus or minus I have to think about what quadrant I’m in. And the problem says theta is between pi over 2 and pi and that means theta is in the second quadrant. And in the second quadrant cosine is negative so cosine is -4/5 because we are in quadrant 2. So now I know the cosine and I have to find the tangent let’s use the tangent identity. Tangent theta equals sine theta over cosine theta.
Sine of theta is 3/5 and the cosine theta is -4/5. So that’s the same as 3/5 times -5 over 4 which is -¾. So for this data where sine is 3/5 cosine is -4/5 and tangent is -¾.