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Problem 31

Solving a Trigonometric Equation In Exercises $31…

Problem 30

Determining a Quadrant In Exercises 29 and 30 , determine the quadrant in which $\theta$ lies.
$$\begin{array}{l}{\text { (a) } \sin \theta>0 \text { and } \cos \theta<0} \\ {\text { (b) } \csc \theta<0 \text { and } \tan \theta>0}\end{array}$$





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Video Transcript

this problem asks us to find the quadrant where the following specified angle exists. So for problem, eh? We need to find the quadrant in which an angle that has signed a two greater than zero and co sent Ada less than zero exists. So to begin, let's find all the places on our unit circle where signed data is greater than zero. So sine theta corresponds to our why values if we look at this, is a cordon of access. So that would mean I'm going to use red. That would mean that sine theta is always greater than zero in this quadrant. Now we also want to find places where co signed data is less than zero. And we find that by looking at our X values on this unit Circle corner axes. So I'm also going to use red for party. But that would be everywhere left who are y axis. So we know that co sign is always negative and quadrants two and three. And this sign is always positive and conscience one into which leaves our answer to be quadrant to where they overlap. Now, the next one gets a little bit trickier because you started throwing in coast. Now remember, the coast seeking data is the same thing as one oversight data. So all the locations where sign is positive. Kosi, can you also be positive? Same goes for their negative values. I'm going to use Green for part B. So if we want Coast Deacon fated to be less than zero, we wanna look for other places where it's negative, which he's going to be. Questions. Three. Before and then Tangent data needs to be greater than zero. So change in data was always positive in quadrants one in three. And you can think about that making sense. Because Tangent Data is signed, data over co signed data. And since Sign and co signer both positive in Quadrant one tangent data is positive as well, and also because sign and co sign our negative in quadrant three. That means the tangent that is going to be positive because a negative number divided by a negative number is a positive number. So we've identified the two regions where both of these statements are true, and we see that they only overlap in quadrant three, which means the cosign see Concetta Cosi contain a sari is less than zero, and tangent data is greater than zero on ly in quadrant three. And so that's the answer to part B. There you go.