Inverse trig functions (arcsin, arccos, arctan)
Inverse Trigonometric Functions
Inverse Trigonometric Functions is a topic in Trigonometric Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(H), which requires students to graph arcsine and arccosine with their domain limitations.
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.