Find two angles in the interval [0°, 360°) that satisfy each o...

Question

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. cos θ = 0.10452846

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine two values of theta that satisfy the given statement, consider values that are in the interval from 0 to 360 degrees inclusive of zero, but not of 360 degrees. Round your answers to the nearest degree. Our given statement is the cosine of theta equals 0.29237166. Our answer choices are answer choice. A 73 degrees and 107 degrees. Answer choice B 61 degrees and 119 degrees. Answer choice C 73 degrees and 287 degrees and answer choice D 61 degrees and 299 degrees. All right. So we're trying to figure out what the value of our angle is what theta is here. And when we have a trigonometric function of an angle equals a number, we can use the inverse function button on our calculators. So because we have cosine here, we'll use the inverse cosine button which is cosine to the negative one like that on a button. And then you want to make sure, very important that you're in degrees for your mode on your calculator. And then you can input that big long number, 0.29237166. And when you do that and hit enter, you should get 73.00000268. And because we're rounding to the nearest degree that rounds to 73 degrees, so there is one answer for our theta. Now, what's another one that's in between zero and 360 degrees. So a couple of ways we can think about this, we can draw 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. Where is that 73 degrees? Well, that's in the first quadrant, we're not up to 90 degrees yet. So we've got the vertex at the origin, initial side along the positive part of the X axis. And then the terminal side is heading up toward positive infinity. For both X and Y, there's our 73 degrees. And one of the ways to think about this is that cosine, we know that that is positive in the first quadrant, all of those trigonometric functions are. So yeah, here we've got cosine equal to a positive number that 0.29 et cetera. But where else is cosine going to be equal to a positive value? We recall from previous lessons that that's going to be in the fourth quadrant. So if we use 73 degrees as our reference angle for the fourth quadrant, we will take 360 degrees minus 73 degrees. And that's going to get us 287 degrees. The other way to think about this is we know that the cosine of X equals the cosine of 360 degrees minus X. And so we have, our other solution is 287 degrees. That one's down in the fourth quadrant. And we look at our answer choices and that matches with answer choice. C well done. We'll catch you on the next one.

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. cos θ = 0.10452846

Key Concepts

Cosine Function and Its Values

Inverse Cosine Function (Arccos)

General Solutions for Trigonometric Equations in [0°, 360°)

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