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Trigonometric Identities - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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