Find the set of values satisfying tan squared 𝜃 plus tan 𝜃 equals zero, where 𝜃 is greater than or equal to zero degrees but less than 180 degrees.
So we’re gonna solve this problem using a method where we’re gonna use the substitution. And the substitution we’re gonna use is that 𝑥 is equal to tan 𝜃. And when we substitute this into our tan squared 𝜃 plus tan 𝜃 equals zero, what we’re gonna get is 𝑥 squared plus 𝑥 equals zero. We see that this is actually something that we can solve quite easily. And we can solve it using factoring, because we can take out 𝑥 as a factor because it’s a factor of both 𝑥 squared and 𝑥. And when we do that, what we’re gonna have is 𝑥 multiplied by 𝑥 plus one equals zero.
So therefore, what we can say is that 𝑥 is gonna be equal to zero or negative one. And the reason we know that 𝑥 is equal to zero or negative one is because we need 𝑥-values that would make the left-hand side equal to zero like the right-hand side. Well, if 𝑥 was zero, then zero multiplied by anything is zero. And if 𝑥 was negative one, then if we look inside the parentheses, we’d have negative one plus one, which would be zero.
Okay, great, so we now know our values for 𝑥. So now if we consider our substitution, what we’re gonna do is substitute back. So therefore, what we can say is that tan 𝜃 also equals zero or negative one because we said that 𝑥 was equal to tan 𝜃. So now what would we do to find out what values of 𝜃 could be?
Well, to work out what the values of 𝜃 are gonna be, what we can do is use the inverse tan function. So we can say that the inverse tan of zero is gonna be equal to. Well, this is just gonna be equal to zero. And we can get that from plugging it into the calculator. Then we can do the same with negative one. And what the inverse tan of negative one is gonna give us is negative 45.
Okay, so we have some principle values for 𝜃. So now what we want to do is consider our interval though because we want to see what values of 𝜃 we’re looking for. Well, we’re looking for the values of 𝜃 that are greater than or equal to zero but less than 180. Well, therefore, from our two principal values, what we can say is that zero degrees is gonna be one of the values that 𝜃 could take. And that’s because we know 𝜃 is greater than or equal to zero.
But now what we want to do is work out what the other possible values are. And to work out all the other possible values, what we’re gonna do is use something called the cast diagram. And the cast diagram is something that looks like this. Well, a cast diagram is divided up to four quadrants. And these quadrants are labeled 𝐴, 𝑆, 𝑇, and 𝐶. Sometimes this is remembered using something like “All Silver Tea Cups.” But what do 𝐴, 𝑆, 𝑇, and 𝐶 mean?
Well, in the quadrant marked 𝐴, it means that all of our values from all of the ratios are going to be positive. In the quadrant marked 𝑆, it’s only gonna be our sine ratio values that are gonna be positive. All the others will be negative. Then for 𝑇, it’s the same for the tangent ratio, and then 𝐶, the cosine ratio. Well, using our cast diagram, we know that one of our principal values is zero degrees. So therefore, we also know that 180 degrees is a possible value. However, we have to disregard 180 degrees because if we look back at our interval, we can see that 𝜃 is greater than or equal to zero but is less than 180 degrees. So it does not include 180 degrees.
Well, we also know that negative 45 degrees was a possible value. But we’d already pointed out this is not within our interval. So how can we use the cast diagram to see which other values this could also give us? Well, if we take a look at what tan negative 45 degrees would be, well, it’d be negative one. So we’re only interested in the quadrants which would give us a negative terms in ratio value. Well, in that case, we can rule out 𝐴 and we can move straight over to 𝑆.
So therefore, what we can do is draw a continuation of the line that’s in the bottom-right quadrant. And what this will give us is an angle of 135 degrees. And that’s because if we go from zero to 90 degrees then we add on 45 degrees, we get 135 degrees. And if you put tan 135 degrees into the calculator, you get negative one. So therefore, what you can say is that the set of values that satisfy tan squared 𝜃 plus tan 𝜃 equals zero, where 𝜃 is greater than or equal to zero but less than 180, are zero and 135 degrees.