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Evaluating Sine and Cosine at Special Acute Angles - 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.

Once you know the unit definitions for sine and cosine, you’re going to want to be able to find values of certain special angles and that’s what we’re going to do right now.

The problem says, find cosine theta and sine theta for theta equals pi over 4. This is one of our special angles. I’ve graphed the unit circle and drawn the angle pi over 4 in standard form. Remember the definition of sine and cosine is x equals the cosine of theta and y equals the sine of theta.

When theta equals pi over 4, we’re going to have x equals cosine pi over 4 and y equals sine pi over 4. Now these two coordinates are on the unit circle and that means they have to satisfy the equation of the unit circle which is x² plus y² equals 1. I don’t have quite enough information to figure out what the sine and cosine are yet but if I notice that pi over 4 is a 45 degree angle and that means that this line, the line that the terminal side is sitting on is y equals x. And that means the x and y coordinates are the same and so this equation becomes x² plus x² equals 1. 2x² equals 1. I divide by 2 and then I take the square root. Plus or minus root ½ which is the same as plus or minus root 2 over 2.

Given that we’re in the first quadrant, the x coordinates is going to be positive so root 2 over 2 and the x coordinate is the cosine of pi over 4, so cosine of pi over 4 is root 2 over 2, the sine of pi over 4 because x an y are the same is also root 2 over 2.

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