Concept Check Suppose that 90°

Question

Concept Check Suppose that 90° < θ < 180° . Find the sign of each function value. tan θ/2

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine whether the value of the following function is positive or negative. If A is between 90 degrees and degrees, our function is the tangent of A divided by three. Our answer choices are answer choice. A positive and answer choice B negative. All right, we've got a few things to figure out here. First, we want to know what our limits are. What angles are we talking about? If A is between 90 degrees and 180 degrees. So we'll plug both of those in for a and see what we get. So we'll have the tangent of instead of a, we'll have the 1st 1 90 degrees divided by three. And then for the second one, we have the tangent of instead of a in the numerator, it's 180 degrees divided by three. Now, we can simplify those. The 1st 1, 90 degrees divided by three is 30 degrees. So we have the tangent of 30 degrees. Two, the tangent of 180 degrees divided by three is degrees. So we're talking about an A any angles from 30 degrees to 60 degrees. We want to know where that is and what's going on with tangent there. So we can draw a rough sketch of a coordinate plane. We've got a vertical Y axis, a horizontal X axis. They come together at the origin in the middle for a 30 degree angle. It's going to have its vertex at the origin and its initial side will be along the positive part of the X axis and then 30 degrees we're heading counterclockwise. So up into the first quadrant and we're heading just a little bit past 30 degrees isn't too big. So it's still closer to the positive part of the X axis than the positive part of the Y axis. So there are 30 degrees. So it's between that and 60 degrees, 60 degrees also has its initial side along the positive part of the X axis vertex at the origin and its terminal side. Still in the first quadrant, this one is closer to the positive part of the Y axis, then the positive part of the X axis. So we're talking about angles in between 30 degrees and degrees and we want to know what tangent is doing there. Well, we recall from previous lessons that we've got a pneumonic device to help us remember what trigonometric functions are positive in which quadrants and really quickly reviewing that the first quadrant is up into the right of the origin. The second quadrant is up into the left of the origin, the third quadrant is down into the left of the origin and the fourth quadrant is down into the right of the origin. Our pneumonic device is all. So we've got a in the first quadrant students. So we've got s in the second quadrant take. So we've got t in the third quadrant and calculus. So we've got c in the fourth quadrant. And since we're talking about the first quadrant here that A in the all standss for a it, excuse me, it stands for all again, because all of our trigonometric functions are positive in the first quadrant. So tangent, yep, tangent is going to be positive in the first quadrant. So if A is between 90 degrees and 180 degrees, the tangent of A divided by three is going to be positive. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Concept Check Suppose that 90° < θ < 180° . Find the sign of each function value. tan θ/2

Key Concepts

Angle Quadrants and Their Ranges

Half-Angle Relationships

Sign of the Tangent Function

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