If n is an integer, n • 180° represents an integer multiple of 1...

Question

If n is an integer, n • 180°  represents an integer multiple of 180°, (2n + 1) • 90°  represents an odd integer multiple of 90° ,  and so on. Determine whether each expression is equal to 0, 1, or  ―1, or is undefined.                                                                                                                                                                                                                                                                                                                                                                                                                                           sin[n • 180°]

Accepted Answer

Welcome back. I am so glad you're here. We're asked to identify if the given expression is undefined, equal to 01 or negative. One note that N is either a positive or negative integer. Our expression is the cosine of degrees plus N multiplied by degrees. Our answer choices are answer choice. A zero answer. Choice B one answer, choice, C negative one and answer choice D undefined. All right, let's choose some values for N, we can start with an N value of one. So now we'll have the cosine of 90 degrees plus one multiplied by degrees. One multiplied by 180 degrees is 180 degrees. So we're taking the cosine of 90 degrees plus 180 degrees. 90 degrees plus 180 degrees is 270 degrees. So we've got the cosine of 270 degrees. We recall from previous lessons and from the tables in the book that degrees is a quadrant angle. And from the tables in the book, the cosine of 270 degrees is zero. All right, let's choose a different value for N let's choose an N value of two. We'd have the cosine of 90 degrees plus two multiplied by 180 degrees. Two multiplied by 180 degrees is 360 degrees. So we have the cosine of 90 degrees plus 360 degrees and degrees plus 360 degrees is going to be degrees. So we're looking at the cosine of degrees. Well, if we want to keep that 450 degrees between zero and 360 degrees, we can take 450 degrees and subtract 360 degrees. And that's going to bring us right back down to 90 degrees. So if we're talking about the cosine of 90 degrees, that's also a quadrant angle. So cosine of 90 degrees is co terminal with here, 90 degrees is co terminal with 450 degrees. And that's also equal to when we, it's the cosine of that 90 degrees, it's also equal to zero. So we've got the cosine of 270 degrees equal to zero and the cosine of 90 degrees both equal to zero. And any value that we choose for N we're going to be co terminal with 90 degrees or degrees. So no matter what we put in for n our answer will be zero. We look at our answer choices and that matches with answer choice. A well, done. We'll catch you on the next one.

If n is an integer, n • 180° represents an integer multiple of 180°, (2n + 1) • 90° represents an odd integer multiple of 90° , and so on. Determine whether each expression is equal to 0, 1, or ―1, or is undefined. sin[n • 180°]

Key Concepts

Sine Function and Periodicity

Integer Multiples of Angles

Trigonometric Values at Key Angles

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