Find all values of θ, if θ is in the interval [0°, 360°) and has ...

Question

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value.

cos θ = -½

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine all possible values of theta in the interval from 0 to 360 degrees inclusive of zero but not degrees which have have the same value as the given function value. Our given function value is the cosine of theta equals negative square at three divided by two. Our answer choices are answer choice. A theta equals 150 degrees and degrees. Answer choice B theta equals 120 degrees and 210 degrees. Answer choice C theta equals 120 degrees and 150 degrees. And answer choice D theta equals 150 degrees and 240 degrees. All right, we recall from previous lessons and from the tables in the textbook that the cosine of theta equals positive square at three divided by 21 theta equals 30 degrees. But here we're talking about when the cosine of data is equal to a negative square at three divided by two. So we need to first figure out in what quadrants is cosine equal to a negative value. And then we can use 30 degrees as our reference angle to figure out what the data will be equal to when cosine equals that negative square at three divided by two. So let's begin with a rough sketch of a coordinate plane. We have a vertical y axis and a horizontal X axis. They come together at the origin in the middle. The first quadrant is up into the right of the origin. Second quadrant up into the left, third quadrant down to the left of the origin and the fourth quadrant down into the right of the origin. And so if we're talking about cosine, we have that pneumonic device to help us remember where the different trigonometric functions are positive. Starting in the first quadrant, we have a then ST see moving around the quadrants in order. And they stand for a for all, all of the trigonometric functions are positive in the first quadrant S stands for sign and its reciprocal function that's in the second quadrant. Those are positive T in the third quadrant stands for tangent and its reciprocal function. And C in the fourth quadrant stands for cosine and its reciprocal function. But that's the one we're looking for here. So we know that cosine is going to be positive in the fourth quadrant and in the first quadrant because they're all positive in the first quadrant. And so, because we're looking for where cosine is negative, we can eliminate the 1st and 4th quadrants. That means we're talking about the 2nd and 3rd quadrants. So we just need to use our reference angle to figure out what the value is going to be for theta in those two. All right. So to figure out a second quadrant angle, we know the reference angle. So we're going to take 180 degrees minus our reference angle, 30 degrees and that equals 150 degrees. So it will be true if the cosine of degrees equals that negative square at three divided by two. Now, for a quadrant three angle, we're going to take 180 degrees plus our reference angle, 30 degrees and 180 degrees plus 30 degrees is 210 degrees. So that one is also true if we plug that in for theta into our cosine of theta equals negative square at three divided by two. And that's as far as we need to go because we need to stay between zero and 360 degrees. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. cos θ = -½

Key Concepts

Unit Circle

Cosine Function

Quadrants of the Coordinate Plane

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