4.6- Inverse Trig Functions

You need to memorize this chart

looks more difficult mentally  but is like

Yea, i know,.. its hard... im sorry

The general rule is, we always use Quadrant I, where cosine, sine, and tangent are all positive. 

Then we want another adjacent quadrant (so the domain can be continuous) that gives us the negatives values. 

Which quadrant to pick is obvious for sine and cosine. 

Sine is still positive in Quadrant II, but negative in Quadrant IV. 
So for inverse sine, we need Quadrant I and Quadrant IV, 
and we have a domain that ranges from -1 to 1. 

Cosine is positive in Quadrant IV, but negative in Quadrant II. 
So for inverse cosine, we need Quadrant I and Quadrant II, 
and we have a domain that ranges from -1 to 1. 

Which quadrant to pick for tangent requires one more consideration. 

From -π/2 to 0 (Quadrant IV), tangent is between -infinity (undefined) and 0. 
From 0 to π/2 (Quadrant I), tangent is between 0 and infinity (undefined). 
From π/2 to π (Quadrant II), tangent is between infinity (undefined) and 0. 

So, for inverse tangent: 
Using Quadrants IV and I results in a continuous domain from -infinity to 0 to infinity. 
Using Quadrants I and II results in a broken domain at the transition from Quadrant I to II. 

So we pick Quadrants IV and I for inverse tangent. 

Summary: 
If the value is positive, inverse sine/cosine/tangent will be in Quadrant I. 
If the value is negative, inverse cosine will be in Quadrant II. 
If the value is negative, inverse sine/tangent will be in Quadrant IV.