When taking cosine of an angle, a result of ½ comes from a 30-60-90 triangle. But which one? Go ahead, draw out the triangle if need be.
Got it? We got the 60° angle, or . This is in Quadrant I, and within inverse cosine's range of [0, π], so everything is golden. Not literally, though, or we would have already sold this stuff off for mad bank.
Example 2
Find cot-1 (1).
To find the inverse of a reciprocal trig function, the easiest thing to do is convert it back to sine, cosine, or, in this case, tangent, by taking the reciprocal of the value. The good news here is that it is really easy because our value is 1, which has a reciprocal of, wait for it, 1.
cot-1 (1) = tan-1 (1)
Now that we are looking for tan-1 (1), we know that we are dealing with a triangle that has two identical sides (how else would a ratio end up as 1?).
That triangle is the 45-45-90 triangle, which means our angle is 45°, or .
Example 3
Find .
When we have a negative angle, we ignore the negative at first. Go away, negative, we don't want to play with you yet. Let's just focus on first.
We know there is a involved in the 30-60-90 triangle, so let's look there a little closer. Whoa, not that close. The numerator in cosine is the adjacent side of an angle, so we want the angle that is adjacent to . That is 30°, or .
This gives us , but not . Cos-1 has a range of 0 to π, which includes Quadrants I and II. Cosine is positive in Quadrant I and negative in Quadrant II. We have a negative angle, so we must be in Quadrant II.
Uh, say wah? How can we get there? It is, like many times before, reference angles to the rescue, riding a white horse and shouting "Find me in Quadrant II." If we look below π (i.e., in Quadrant II), we find an angle of . This is the angle that will give us .
. This is in Quadrant I, and within inverse cosine's range of [0, π], so everything is golden. Not literally, though, or we would have already sold this stuff off for mad bank.
.
.
first.
involved in the 30-60-90 triangle, so let's look there a little closer. Whoa, not that close. The numerator in cosine is the adjacent side of an angle, so we want the angle that is adjacent to 
. That is 30°, or 
.
, but not 
. Cos-1 has a range of 0 to π, which includes Quadrants I and II. Cosine is positive in Quadrant I and negative in Quadrant II. We have a negative angle, so we must be in Quadrant II.
below π (i.e., in Quadrant II), we find an angle of 
. This is the angle that will give us 
.