Concept Check Suppose that 90°

Question

Concept Check Suppose that 90° < θ < 180° . Find the sign of each function value. cos ( ―θ)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine whether the value of the following function is positive or negative. If B is between 90 to degrees, our function is the C can of negative B. Our answer choices are answer choice. A positive and answer choice B negative. All right. So we have a couple of different things that we need to figure out here. First, we're told that B is between 90 to 180 degrees. So let's figure out what the limits of B are. If we plug 9180 degrees in. For B first, we're taking the C can of negative and instead of B, we've got 90 degrees and then we're taking C can of negative and instead of B, we have 180 degrees and so negative 90 degrees and negative 180 degrees. We can figure out where those are on a coordinate plane. We can draw a rough sketch. We've got a vertical Y axis, a horizontal X axis. They come together at the origin in the middle, we have four quadrants up into the right of the origin is the first quadrant up into the left of the origin is the second quadrant down into the left of the origin is the third quadrant and down into the right of the origin is the fourth quadrant. And so we're talking about angles of negative 90 degrees and negative 180 degrees. Where are those on our coordinate plane? Well, they're both quadrant angles. We know that we'll have an initial side along the positive part of the X axis for both of these. So a vertex of the origin heading out to positive infinity for the X values for the initial side and for negative degrees because it's negative, we're going to head clockwise from our initial side instead of counterclockwise as we would for a positive angle. So our negative 90 degrees is going to have its terminal side along the negative part of the Y axis. That's our negative 90 degrees. And so it's between that and now let's find negative degrees. All right, same thing with its initial side along the positive part of the X axis. Negative. Again, means we're heading clockwise from there and our terminal side at negative 180 degrees. That's going to be along the negative part of the X axis, just like positive 180 degrees would be. But the direction is important here because we know that it's going to be between negative 90 degrees and negative 180 degrees which tells us that we're in the third quadrant. So now we ask ourselves, well, what is C can't in the third quadrant? Is it positive or negative? And we have a pneumonic device to help us remember where the different trigonometric functions are positive and negative and that pneumonic device is all. So a that's in the first quadrant students. So s that's in the second quadrant take. So T that's in the third quadrant and calculus. So C that's in the fourth quadrant. And so A for all that still stands for all because all of the trigonometric functions are positive in the first quadrant. S for students that stands for sign in the second quadrant. And it also includes its reciprocal function Cosic can t for take in the third quadrant that stands for tangent as well as its reciprocal function cotangent and C for calculus in the fourth quadrant that stands for cosine and also includes its reciprocal function C can, there's C can, that's the one we're talking about. So we know that it's positive in the fourth quadrant C can is also going to be positive in the first quadrant because they all are. So what's doing in the third quadrant? Well, it's going to be negative C can, is negative in the third quadrant. Therefore, if B is between 9180 degrees, the C can of negative B is going to be a negative value. So we look at our answer choices and that matches with answer choice. B well done. We'll catch you on the next one.

Concept Check Suppose that 90° < θ < 180° . Find the sign of each function value. cos ( ―θ)

Key Concepts

Quadrants and Angle Ranges

Even-Odd Identities in Trigonometry

Sign of Cosine Function in Different Quadrants

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