Introduction
An inverse function undoes an operation. If , then . But wait! as well. Which angle should the calculator give back?
To make the inverse a real function (one output), we have to restrict the range of possible answers.
Prerequisite Connection
You know the six trig functions and their graphs. Now we need to "undo" them by defining inverse functions.
Today's Increment
We're adding Inverse Trig Functions with specific domain restrictions to make them pass the Vertical Line Test.
Why This Matters
In Calculus, definite integrals involving square roots often simplify to inverse trig functions: .
Inverse Sine (Arcsin)
We restrict Sine to quadrants I and IV.
Range: .
Inverse Cosine (Arccos)
We restrict Cosine to quadrants I and II.
Range: .
Inverse Tangent (Arctan)
Restricted to open interval .
It has horizontal asymptotes at .
Arcsecant and Arccosecant
Arcsecant
Range: .
Arccosecant
Range: .
Worked Examples
Example 1: Evaluating Inverses
Evaluate .
Identify Reference Angle
We know .
Apply Restriction
Since the input is negative, and sine is restricted to , we must go "down" to Quadrant IV.
Answer: . (NOT ).
Example 2: Composition
Evaluate .
Draw a Triangle
Let . This means .
Opposite = 3, Hypotenuse = 5.
Find Adjacent Side
Pythagorean Triple! 3-4-5. The adjacent side is 4.
Calculate Cosine
.
Example 3: Variable Expressions (Advanced)
Write as an algebraic expression.
Set up Triangle
.
Adjacent = x, Hypotenuse = 1.
Find Opposite Side
.
.
Evaluate Tangent
.
Practice Quiz
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