In Exercises 17–22, let θ be an angle in standard position. Name ...

Question

In Exercises 17–22, let θ be an angle in standard position. Name the quadrant in which θ lies.

tan θ < 0, cos θ < 0

Accepted Answer

Hello everybody. Now we're doing all right. Today, today we're going to look at this question that states write the quadrant in which data lies and consider the angle theta in the standard precision. Only. Now they let us know that tangent data is less than zero and cosine data is greater than zero. And they give us a couple of answers with answer choice A being quadrant one answer choice B being quadrant two, answer choice C being quadrant three and answer choice D being quadrant four. Now let's know the information that they gave us. So they tell us that cosine theta is greater than zero. So let's know when is cosine theta positive. So cos data is positive. So when you think about the, think about X values, so when are X values positive? Well, X values are positive to the right of the Y axis. So those quadrants will be quadrant one and quadrant four. So cosine data is positive in quadrants one and four. So we know we're gonna be dealing with either quadrant one or quadrant four. Now, we're looking at tangent data, it's a bit more complicated. So we'll see that tangent data is sine theta divided by cosine theta. Therefore, we need to find that signed data and cosine data must be opposite. They must be opposite signs for tangents data two B negative. And we want it to be negative. We told because they told us that tangent data is less than zero, which is negative. So we need to find quadrants that sine data and cosine data are opposite in sine. So let's make notes about each sign and cosine, well, Sine theta, when we look at sin theta, we want to think about Y values instead of X values. So when is, when are Y values positive? Well, they're positive above the X axis. So assign data is positive in quadrants. So what quadrant are above the X axis in one and two? So that's where sine data will be positive and it will be negative in quadrants. Now, we look at the quadrants below the X axis because that's where Y values are negative and those will be quadrants three and four. And that is it for sign. So now we look at cosine data. So cosine theta is positive and like I said, well, we already noted where cosine theta is positive, cosine theta is positive in quadrants one and four and it is negative in quadrants. Well, now we have to see where X values are negative, they're negative to the left of the Y axis. So those are quadrants two and three. So now what we can do is eliminate the quadrants where sine and cosine have the same sign. So when, where they both positive, well, we have quadrant one as positive in sine and also positive in cosine. So we can eliminate quadrant one and where, where are they both negative? We see that they share quadrant three in their negative values. So we're left with what we're left with that tangent data is negative and quadrants two and four because those are the quadrants where sine and cosine have the opposite sign one is positive and the other one is negative. Now, we mentioned earlier that we are dealing with quadrants one and four because cosine theta is positive in one and four. Now if we compare that to our tangent values, we have two and four. So we see that they have the fourth quadrants in common, which means that our data will lie in that quadrant that they share in common or answer choice D quadrant number four. So I hope that was helpful. And until next time.

In Exercises 17–22, let θ be an angle in standard position. Name the quadrant in which θ lies. tan θ < 0, cos θ < 0

Key Concepts

Quadrants of the Coordinate Plane

Trigonometric Functions and Their Signs

Analyzing Inequalities in Trigonometry

Watch next