Inverse Trigonometric Functions

Inverse trigonometric functions undo the trigonometric functions: given a value, they return the angle that produced it. Because sine, cosine, and the others are not one-to-one on the whole real line, each inverse function uses a carefully chosen principal value interval.

What You Will Learn

• Why Principal Values — why each inverse function restricts its range, and what interval is chosen for each one.
• The Six Inverses — arcsine, arccosine, arctangent, arccotangent, arcsecant, and arccosecant: domains, ranges, monotonicity, and graphs.
• Evaluating — how to read special-angle values such as $\arcsin \frac{\sqrt{3}}{2} = \frac{\pi}{3}$ using the principal interval.
• History — how the “arc” notation developed from solving trigonometric equations and integration problems.

Chapter Sections

• Arcsine Function
• Arccosine Function
• Arctangent Function
• Arccotangent Function
• Arcsecant Function
• Inverse Cosecant Function

Start with Arcsine Function to see the complete pattern, then the other five follow the same structure.