Evaluating trigonometric functions Without using a calculator,...

Question

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

cos (2π/3)

Accepted Answer

Welcome back, everyone. In this problem, we want to solve for the exact value of the trigonometric expression of the cosine of 3 fourths of pi. For our answer choices, a is the negative square root of 3 divided by 2. B is the square root of 3 divided by 2. C is the negative square root of 2 divided by 2. And the d is the square root of 2 divided by 2. Now, we want to find the exact value for our trigonometric expression. And since our angle is less than 2 pi, we can represent it easily on our unit circle. So, if I go to the right here and I draw my unit circle, something that looks like this. Okay? And we have our y axis and our x axis. 3 fourths of pi on our unit circle would probably be somewhere over here. Okay? And we want to figure out the exact value of whatever that is. Now, we don't know what 3 fourths of pi is for cosine but there's an property that we can use for the cosine because recall that for the cosine of an angle, the cosine of pi minus x radians equal the negative value of the cosine of x. In other words, if we have an angle that's less than pi radians, then we can use this property to help us figure out what its reference angle would be. Now, let's think about it. We have angle 3 fourths of pie. So, the question is, what minus or in other words, pie minus what would give us 3 fourths of pie? Well, if we think about it, that value would have to be pi minus 3 fourths of pi. And pi, which can be written as 4 fourths of pi, would mean 4 pi minus 3 pi all divided by 4, which would be a 4th of Pi. So that's our value of x. So this tells us then that the cosine of 3 4ths of Pi can be rewritten using this property as the negative value of the cosine of a 4th of pi. Do we know what the cosine of a 4th of pi is? Yes. We do. Because, recall that the cosine of a 4th of pi equals the square root of 2 divided by 2. So now by substituting it in our problem, that means the cosine of 3 fourths of pi can be written as the negative value of the square root of 2 divided by 2. When we look back in our answer choices, that would have been answer choice c. Thanks a lot for watching everyone. I hope this video helped.

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

cos (2π/3)

Key Concepts

Unit Circle

Reference Angles

Cosine Function

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