Inverse trig functions (arcsin, arccos, arctan)
Inverse Trigonometric Functions
Inverse Trigonometric Functions is a topic in Trigonometric Functions in the Common Core State Standards. It is aligned to Standard F-TF.6, F-TF.7, which requires students to understand and use the inverse trigonometric functions.
The inverse trigonometric functions \(\arcsin\), \(\arccos\), and \(\arctan\) undo the trigonometric functions on restricted domains, returning a single principal-value angle.
Theory
Trig functions repeat, so they are not one-to-one — to invert them we restrict the domain to a piece that is. The inverses return a single principal value:
- \(\arcsin x\): range \(\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\).
- \(\arccos x\): range \([0,\pi]\).
- \(\arctan x\): range \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\).
Each answers “which angle (in the allowed range) has this sine / cosine / tangent?”
The defining ranges:
How to evaluate an inverse trig expression
- Rephrase as “which angle has this value?”
- Restrict the answer to the function's range.
- Use reference angles and the correct quadrant for the range.
- For compositions, introduce an angle for the inner inverse and build a triangle.
Ask: which angle in \(\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\) has sine \(\dfrac12\)?
| \(\arcsin\dfrac{1}{2}\) | \(=\) | \(\dfrac{\pi}{6}\) |
Arccosine outputs values in \([0,\pi]\); a negative input lands in Quadrant II.
| \(\arccos\!\left(-\dfrac{\sqrt2}{2}\right)\) | \(=\) | \(\dfrac{3\pi}{4}\) |
Which angle in \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\) has tangent \(1\)?
| \(\arctan 1\) | \(=\) | \(\dfrac{\pi}{4}\) |
Let \(\theta=\arccos\dfrac35\), so \(\cos\theta=\dfrac35\) with \(\theta\) in \([0,\pi]\). Build a right triangle: adjacent 3, hypotenuse 5, so opposite \(=4\).
| \(\sin\theta\) | \(=\) | \(\dfrac{4}{5}\) |
Common pitfalls
Frequently asked questions
What is arcsin?
The inverse sine function: \(\arcsin x\) returns the angle in \(\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\) whose sine is \(x\).
Why do inverse trig functions have restricted ranges?
Because the trig functions repeat and are not one-to-one. Restricting the domain makes each invertible and gives one principal value.
Is sin to the minus 1 the same as cosecant?
No. \(\sin^{-1}x\) is arcsine, the inverse function; \(\csc x=\dfrac{1}{\sin x}\) is the reciprocal. They are different.
How do you evaluate something like sin(arccos x)?
Let the inner inverse be an angle, draw a right triangle with the given ratio, find the missing side, and read off the outer function.