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 = csc⁻¹ (-2)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the angle measure theta without using a calculator express the answer in degrees. Then we're given that theta equals the inverse cosecant of negative two square root three divided by three. Our answer choices are answer choice, a 150 degrees. Answer choice B 60 degrees, answer choice C 240 degrees and answer choice D negative 60 degrees. All right. So we're talking about the inverse kant. And the first thing we need to do is figure out if negative two square root three divided by three is within the domain of our inverse KC camp. So we recall from previous lessons and looking at tables in the textbook and we can see that the domain for inverse kosic can is from negative infinity to negative one inclusive of negative one in union with one to infinity. Now, if we were to figure out what negative two square root three divided by three is that is a proximately negative 1.15 and that keeps going. But that's enough for us to see that it is within our domain. It's in the part where from negative infinity to negative one. So we can proceed. The next thing we want to do is kind of reframe this, we know that theta equals the inverse kant of negative two square root three divided by three is the same as theta is the angle whose kant is negative two square root three divided by three. Or in other words, we can rewrite this kant of theta equals two square root three divided by three, which is great because this is one of those special angles that we've got the tables in the textbook to tell us. Oh, whoops, sorry, I should have written negative two square root three divided by three. And anyway, we've got those tables in the textbook to tell us what the cosecant of theta is when it's equal to a positive to square root three divided by three. Like I was writing when that's the case theta is equal to 60 degrees. So just figuring out for the Kent of theta equals a negative two square root three divided by three. Let's think about a coordinate plane. We've got a vertical Y axis, a horizontal X axis, they come together the origin. And when we're talking about Ko camp being equal to a negative two square root three divided by three kant is going to be negative in the third quadrant and the fourth quadrant. However, it's important to note we're talking about the inverse Kam which is a 1 to 1 function and so our range is going to be restricted here, we can't have any repeating why values. And so the range again, going from those tables in the textbook and previous lessons, the range for inverse Kan is from including negative pi divided by 2 to 0, but not including zero. So we put parentheses in union with starting right after zero again, 02 pi divided by two and ending with a bracket. But since we're talking about degrees here, that would be inclusive of negative 90 degrees to zero. But parentheses at zero in union with parentheses, zero to positive 90 degrees closing with the bracket. So looking at our coordinate plane, the negative 990 degrees is the negative part of the Y axis all the way up to the X axis, the positive part, but not including that. And then from the positive part of the X axis, but not including that all the way to the positive part of the Y axis. So if that's our range, we can eliminate all of those negative kant function or angle measures from the third quadrant because we're only talking about the 1st and 4th quadrants. So for our negative, that's going to be in the fourth quadrant. Now our reference angle, we said Koi can of theta equals a positive two square root three divided by three when theta equals 60 degrees. And here because our ranges from negative 90 not including zero up to positive 90 our fourth quadrant angle is going to be a negative 60 degrees to remain within that range. So we know that theta equals negative 60 degrees. When the inverse kant is, when we're taking the inverse kant of negative two square root three, divided by three, we look at our answer choices and that 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 = csc⁻¹ (-2)

Key Concepts

Inverse Cosecant Function (csc⁻¹)

Domain and Range of csc⁻¹

Evaluating Trigonometric Values Without a Calculator

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