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 = arccos (-1/2)

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're given the trigonometric equation that theta equals the inverse cosine of negative square root three divided by two. Our answer choices are answer choice, a 60 degrees, answer choice B 30 degrees, answer choice C 210 degrees and answer choice D 150 degrees. All right. The first thing we need to do is figure out whether negative square at three divided by two is within the domain of our inverse cosine function. And so we recall from previous lessons and from the tables in the textbook that the domain of the inverse cosine function is from, inclusive of so with a bracket, negative one to positive one and with a bracket and the negative square root of three divided by two is approximately negative 0.866. And then it keeps going. But we can see that that is between negative one and one. So we can proceed. So the next thing we want to do is rephrase this. So saying that theta equals the inverse cosine of negative square root three divided by two is the same as saying that the cosine of theta equals negative square root three divided by two. And that's a little bit easier to think of because now we just need to figure out what is our angle measure when cosine equals negative square root three divided by two. And we can recall from previous lessons and from tables in the textbook that the cosine of theta equals a positive square at three divided by two. This is one of those special angles when theta equals 30 degrees. So we just need to figure out when that's true for cosine of theta equals a negative square root three divided by two. Now a couple of things to note here because we're talking about inverse cosine, this is a 1 to 1 function. And so we have a restricted range here as well. We recall from previous lessons and from the tables in the textbook that the range of our inverse cosine function is from zero to pi inclusive of both. And since we're told to express our answer in degrees, thinking about this in degrees, that's from zero degrees to 180 degrees. Now, if we draw a quick sketch of a coordinate plane, we have a vertical y axis, a horizontal x axis, those come together at the origin, our cosine function is negative just like we are given here, cosine of theta equals a negative square root three divided by two in the 2nd and 3rd quadrants. So we're looking in the 2nd and 3rd quadrants. However, the range of our inverse cosine function is from zero, the positive part of the X axis to 180 degrees, the negative part of the X axis. So we can exclude any values from the third quadrant. And we're only talking about the second quadrant. Now, our reference angle from the first quadrant, we said that the cosine of theta equals a positive square root three divided by two when theta equals 30 degrees. So if our reference angle in quadrant one is 30 degrees, what's going to be our quadrant two angle? Well, that's 180 degrees minus 30 degrees or 150 degrees. So theta equals 150 degrees when the cosine of theta equals a negative square root three divided by two. When we're talking about theta equals the inverse cosine of negative square root three divided by two, we look at our answer choices and this matches with answer choice. D 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 = arccos (-1/2)

Key Concepts

Inverse Trigonometric Functions

Cosine Values of Special Angles

Unit Circle and Angle Quadrants

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