The Inverse Cosine and Inverse Tangent Functions
Simple Explanation
arccos(x) gives the one angle in the range [0,Ο] whose cosine is x (domain [β1,1]). arctan(x) gives the one angle in the range (βΟ/2,Ο/2) whose tangent is x β and since tangent covers all real numbers, arctan's domain is all real numbers too.
Why Do We Need It?
Each inverse trig function needs its OWN specific restricted range to be well-defined β arccos's range differs from arcsin's, since cosine and sine repeat their values in different patterns.
See It
An S-shaped curve defined for all real x, leveling off toward horizontal asymptotes at y = pi/2 and y = -pi/2
Formula
The Inverse Cosine and Inverse Tangent Functions
arccos(x): domain [β1,1], range [0,Ο]. arctan(x): domain all reals, range (βΟ/2,Ο/2)
arccos(x) gives the one angle in [0,Ο] whose cosine is x; arctan(x) gives the one angle in (βΟ/2,Ο/2) whose tangent is x.
- x
- β the known cosine value (for arccos) or tangent value (for arctan)
When to use it: Whenever an angle needs to be recovered from a known cosine or tangent value.
Worked Example
Evaluate inverse cosine and inverse tangent
Find arccos(0.5) and arctan(1), both in degrees.
Why Does This Work?
Just as with arcsin, each of these functions needs a restricted range containing exactly one angle for every valid input β arccos uses [0,Ο] (matching cosine's natural decreasing pattern there) and arctan uses (βΟ/2,Ο/2) (matching one full branch of tangent's repeating pattern).
Real-Life Example
Finding a heading angle from velocity components
A navigator knows a ship's northward and eastward speed components and needs the actual compass heading angle.
arctan of the ratio of these two components gives the heading angle directly.
Practice
Find arccos(0) in degrees.
MediumCommon mistake
Confusing arccos's range ([0Β°,180Β°]) with arcsin's range ([β90Β°,90Β°]) β each inverse trig function has its own specific restricted range.
Quick Review
- arccos(x): domain [β1,1], range [0,Ο].
- arctan(x): domain all reals, range (βΟ/2,Ο/2).
- Each inverse function has a DIFFERENT restricted range, matched to its own function's pattern.