Find the cot of 𝜃 given the sin of 𝜃 is three-fifths, where 𝜃 is between 90 degrees and 180 degrees.
So we are trying to find the cot of 𝜃. And using trig identities, we know a few things. We know that cot of 𝜃 is equal to the cos of 𝜃 divided by the sin of 𝜃, which is actually the exact same thing as cot of 𝜃 is equal to one over tan of 𝜃 because tan of 𝜃 is equal to the sin of 𝜃 divided by the cos of 𝜃. So when you take one divided by that, it just flips it, cosine over sine. Now this isn’t going to be very helpful because we know sine, however we don’t have any information about cosine. So we don’t wanna use these.
Now, our second equation, we have cot squared 𝜃 plus one equals csc squared 𝜃. And csc 𝜃 is equal to one over sin 𝜃, and we know sine. So that would only leave cot squared 𝜃 left, and we could square root and find just the cot of 𝜃. So let’s go ahead and use this equation.
So as we said, we can replace csc 𝜃 with one over sin 𝜃. And then of course, we have to square it. So now let’s replace sin of 𝜃 with three-fifths, so that means we have one divided by three-fifths on the inside of the parenthesis. Now, one is the same as one over one, so if we’re taking one over one and dividing by three-fifths, that means we’re actually going to end up multiplying by the denominator’s reciprocal. So essentially, we flip the bottom of the fraction. And now we multiply straight across. So one times five is five, and one times three is three. So we get five-thirds. And if we square five-thirds, five squared is 25 and three squared is nine.
So now our next step would be to isolate cot squared 𝜃. So let’s subtract one from both sides. The ones on the left cancel but on the right, twenty-five ninths minus one, we can make one be nine over nine because that’s equivalent to one. Now when we subtract fractions, we subtract the numerators and we keep the denominators. So we have cot squared 𝜃 is equal to sixteen-ninths. Now to solve for cotangent, we need to square root both sides. So the square root of 16 is four and the square root of nine is three. So the cot of 𝜃 is equal to four-thirds.
Now it also gives us another piece of information, that 𝜃 is between 90 degrees and 180 degrees. So looking at our graph here, we have zero degrees then 90, 180, 270. And then actually, when we make a whole circle, we end up at 360 degrees which is the exact same as zero degrees. So it says 𝜃 is between 90 and 180, so that means 𝜃 is in here. Now, we’re finding cotangent and it’s important to know that we’re looking at a graph, 𝑥 represents the cos of 𝜃 and 𝑦 represents the sin of 𝜃. And cotangent is equal to cos of 𝜃 divided by sin of 𝜃. So we’re only looking at this to decide if it’s positive or negative, the four-thirds.
So in quadrant number two, out of all four quadrants, we’re looking at quadrant two for 𝜃 because we’re between 90 degrees and 180 degrees. Here, we can see that 𝑦 is positive but 𝑥 would be negative. So that means that sine is positive, and cosine is negative. And a negative divided by a positive is a negative. So we need to make our four-thirds a negative four-thirds because of where 𝜃 is located.
So the cotangent of 𝜃 is equal to negative four-thirds.