You know what would make us really happy? Jetpacks. We'd fly around fighting crime, and bears, and criminal bears. Woosh. Like a superhero, but Shmoopier.
Anyway, that's pretty tangential to what we're here to talk about, which is the tangent of different angles. We've also got cosecant, secant, and cotangent to do as well. Luckily, they all work the same way. If they worked different ways, that'd be four times the work, and we'd honestly flake out before finishing.
The key to finding angles for these functions is to know that they're all knock-off brand soda to sine and cosine's Coke and Pepsi.
How about we solve a problem? We're thinking…
.
The angle is a little less than π, so it's in the second quadrant.
That's our corresponding acute angle. We need to find its cotangent. Looking back at that list we made just a minute ago, we see that:
We stare at our left hand for a little bit, and discover that cos is 
and sin is 
.
That means the value of cot is 
, but now we need its sign. We're dividing cosine by sine, so in the second quadrant we're dividing a negative (cosine) by a positive (sine). If trig functions were magnets, we'd have a hard time pulling them apart now. They aren't, though, so instead we have:
In order for cotangent (or tangent) to be positive, both cosine and sine must be either positive or negative together, like in Quadrants I and III. Quadrants II and IV are negative. Quadrant V exists outside of our dimension, and constantly produces ice cream and volcanoes. They tend to balance each other out.
Cosecant and secant have the same sign as sine and cosine, respectively. We don't need a whole new sign language to work with them.
There is a danger to finding the angles of trig functions. All four trig functions other than sine or cosine involve division, so there is the risk of the ever-dreaded division by zero. Tangent, for instance, would divide by zero whenever cosine equals zero, at 0 and π radians. At those spots, the function is undefined, like the sound of one hand clapping.
?
?
, in the third quadrant.
?
smaller than π, so it's in the second quadrant.
.
?
in the third quadrant.
?
in the first quadrant.
?
in the fourth quadrant.
