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 = arcsin (-√3/2)

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 sign of negative one half. Our answer choices are answer choice, a negative 30 degrees, answer choice B 210 degrees, answer choice C 150 degrees and answer choice D negative 60 degrees. All right. So in order to determine the here, the first thing we need to do is figure out if our negative one half is within the domain of the inverse sine function. So here we've got previous lessons and the tables in the textbook that tell us what the domain of the inverse sine function is. And that is from inclusive of. So we're using brackets negative one to positive one and yeah, negative one half is within that domain. So we can proceed next. We're going to think about this. We can rephrase it saying that theta equals the inverse sign of negative one half is the same as saying that theta is the angle whose sign is negative one half. Or in other words, the sign of theta equals negative one half. And we do have a limit here for the range of our theta values. That can be the answer because we're talking about an inverse function. So we have a limited range. The range here has to be from inclusive of negative pi divided by two to positive pi divided by two. That's from the tables in the textbook and previous lessons or since we're talking about degrees here, that's from negative 90 degrees to positive 90 degrees. So our angle needs to be within that. But we can look at those previous lessons and the tables in the textbook for when is the sign of theta equal to negative one half? While the tables tell us that the sign of theta equals a positive one half when theta equals 30 degrees. And if we draw a really quick sketch of a coordinate plane, we have a vertical Y axis, a horizontal X axis. Though those come together at the origin, we have 30 degrees in quadrant one where our sine function is positive. But we're talking about a negative value for our s so it's negative one half sine is negative in quadrants three and four. So we're talking about quadrants three and four here. But again, our range is limited from negative 90 degrees. That's the negative part of the Y axis to positive 90 degrees, the positive part of the Y axis. So we can exclude any values any angles from quadrant three. We're just talking about quadrant four here. So what is the angle that we're talking about in quadrant four? If our reference angle is going to be 30 degrees in the first quadrant, well, that's going to be staying within our range a negative 30 degrees. So this is true sign of theta equals negative one half. When theta equals negative 30 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 the degree measure of θ if it exists. Do not use a calculator.
θ = arcsin (-√3/2)

Key Concepts

Inverse Sine Function (arcsin)

Sine Values of Special Angles

Sign and Quadrant Considerations for arcsin

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