Find the degree measure of θ if it exists. Do not use a c...

Question

Find the degree measure of θ if it exists. Do not use a calculator.

θ = cot⁻¹ (-√3/3)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the angle measure theta without using a calculator express the answer in degrees. Then we are given that theta equals the arc cotangent of negative one. Our answer choices are answer choice, A 135 degrees. Answer choice B 315 degrees. Answer choice C negative 30 degrees and answer choice D 120 degrees. All right. So we are trying to determine the inverse cotangent of negative one. And the first thing we need to do is figure out if negative one is within the domain of our inverse or our cotangent, we recall from previous lessons and from the tables in the textbook that the domain of the inverse cotangent is from negative infinity to positive infinity. And yeah, negative one does fall within that. So we can proceed. Now, the next thing we want to do is kind of reframe this. We know that when we have theta equals the arc cotangent of negative one, it's the same as asking theta is the angle whose cotangent is negative one. Or in other words, we can say the cotangent of theta equals negative one. And that makes it a little bit easier for us to solve for theta. We recall from previous lessons in the tables in the textbook that when we have the cotangent of an angle, theta equal to a positive one, that's true when theta equals 45 degrees. However, we're not strictly talking about this. We're, we are talking about the inverse cotangent and that has a restricted range. We recall from previous lessons in the tables in the textbook that the range of our cotangent is from not inclusives who use parentheses from zero to pi or since we're expressing our answer in degrees from zero degrees to 180 degrees. And so that means if we draw a quick 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 range of our acceptable answers are from zero degrees, the positive part of the X axis to 180 degrees, the negative part of the X axis. So if we have a cotangent equal to a negative one, where is cotangent going to be negative cotangent is negative in quadrants two and four. But we've already said quadrant four is outside of our range. So we can exclude any answers from quadrant four. We're just looking at quadrant two, if 45 degrees is our reference angle. We said that the cotangent of theta equals positive one when theta equals 45 degrees. If that's our reference angle, what's our angle in quadrant two? Well, that's going to be 180 degrees minus 45 degrees, which gives us 135 degrees. So our theta equals 135 degrees. When the arc, the arc cotangent equals negative one, we look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Find the degree measure of θ if it exists. Do not use a calculator.
θ = cot⁻¹ (-√3/3)

Key Concepts

Inverse Trigonometric Functions

Cotangent Function

Reference Angles and Quadrants

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