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 = arctan (-1)

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 it in degrees. Our given trigonometric function is that theta equals the arc tangent of the square root of three divided by three. Our answer choices are answer choice, a 60 degrees, answer choice B 30 degrees, answer choice C negative 30 degrees and answer choice D negative 45 degrees. All right. So we're looking at the arc tangent or the inverse tangent of the square root of three divided by three. And the first question we ask ourselves is, is the square root of three divided by three within the domain for inverse or arc tangent. Well, here we can recall from previous lessons and from the tables in the textbook where it tells us the domain of the inverse tangent which is from negative infinity to positive infinity. And yeah, the square root of three divided by three does fall within that domain. So we're good to proceed. Now, it can help to rethink of our given trigonometric function. And that is the same as saying that theta is the angle whose tangent is the square root of three divided by three. Or in other words, we can rewrite this as the tangent of theta equals square of three divided by three. Now, the only thing here is that we do have a restricted range for our angle theta, we can look at those same tables in the textbook and recall from previous lessons that our range is from negative pi divided by two to positive pi divided by two. or since we're working in angle or degree angles here, that's from negative 90 degrees to positive 90 degrees. And so if we think about our angles, we do have previous lessons and those tables in the textbook that tell us exactly what angle measure the tangent of theta is for square root three divided by three. That's one of those special angles and looking at those charts that is true when theta equals 30 degrees. Now, normally, when we'd be solving for that, we'd be saying OK, which, which quadrants are we in? And if we draw a quick coordinate plane, we have a vertical y axis, a horizontal X axis, they come together the origin in the middle and tangent is positive. So it's square root three divided by three that's positive in the first quadrant and the third quadrant. However, our range here is only from negative 90 degrees to positive 90 degrees. We're talking about only the first quadrant and the fourth quadrant. So here we only need to have that one answer where theta equals 30 degrees. So that's our answer. We look at our answer choices and that matches with answer choice. B 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 = arctan (-1)

Key Concepts

Inverse Trigonometric Functions

Tangent Function

Quadrants of the Unit Circle

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