Trig Functions with the Unit Circle Exercise 6

Answer:  \(\theta=225^\circ\)

Solution Method: In this problem, we need to find \(0^\circ\) \(\leq \theta \leq\) \(360^\circ\) satisfying two conditions. First, let’s narrow down the possible quadrants for \(\theta\).

\(\tan\theta>0\) means tangent needs to be positive. Using our acronym ASTC, we know that tangent is positive in Q1 and Q3.

We also have that secant is negative. Secant is the reciprocal of cosine, so their signs will be the same. So we need cosine to be negative, which occurs in Q2 and Q3.

Now \(\theta\) has to satisfy BOTH conditions because it’s an "and" statement. So \(\theta\) can only be in Q3.

Now that we have the quadrant, we need to determine if \(\theta\) has a reference angle of \(30^\circ,\) \(45^\circ,\) or \(60^\circ.\) I don’t know the secant values on the unit circle off the top of my head, but I do know the cosine values because I know the x-coordinates. If \(\sec{\theta}=-\sqrt{2},\) \(\cos{\theta}=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}\). Reference angles of \(45^\circ\) have x-coordinates of \(\pm\frac{\sqrt{2}}{2}\).

So then \(\theta\) is in Q3 and has a reference angle of \(45^\circ,\) so \(\theta=225^\circ\) is the only answer.