Inverse Trigonometric Functions
In short: Inverse trigonometric functions such as arcsin, arccos and arctan take a ratio and return the angle that produces it, with the answer restricted to one standard interval.
PrecalculusTrigonometryF-TF.B.6
What is Inverse Trigonometric Functions?
The inverse trigonometric functions answer a backward question: which angle has this sine, cosine or tangent? They are written \(\sin^{-1}x\) (arcsin), \(\cos^{-1}x\) (arccos) and \(\tan^{-1}x\) (arctan), where the “-1” marks an inverse, not a reciprocal.
Because sine, cosine and tangent repeat, each must be restricted to a one-to-one piece first. The outputs are: \(\sin^{-1}x\) in \([-\frac{\pi}{2}, \frac{\pi}{2}]\), \(\cos^{-1}x\) in \([0, \pi]\), and \(\tan^{-1}x\) in \((-\frac{\pi}{2}, \frac{\pi}{2})\). Only inputs from -1 to 1 are allowed for arcsin and arccos.
How to spot it
- A right triangle has two known sides and the angle is unknown, so a calculator key like sin^-1 is required.
- A trig equation such as sin x = 0.4 needs a solution angle, which the inverse trigonometric functions supply, in a stated interval.
- A problem asks for an exact angle in radians like arccos(-1/2) without a calculator.
- An expression nests a trig function inside its inverse, such as sin(arccos(3/5)).
The method
- Rewrite the question as an equation: \(y = \sin^{-1}x\) means \(\sin y = x\) with \(y\) in the allowed range.
- Find the reference angle whose sine, cosine or tangent has the size of \(x\).
- Place the angle in the correct quadrant for the restricted range, using the sign of \(x\).
- Check by evaluating the original trig function at your answer; inverse trigonometric functions must reproduce the input.
Inverse Trigonometric Functions: solved example
Problem
Evaluate \(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\), \(\cos^{-1}\left(-\frac{1}{2}\right)\) and \(\tan^{-1}(-1)\) in radians.
Solution
For arcsin: sine equals \(\frac{\sqrt{3}}{2}\) at \(\frac{\pi}{3}\); a negative input pushes the angle below zero within the range, giving \(-\frac{\pi}{3}\). For arccos: cosine equals \(\frac{1}{2}\) at \(\frac{\pi}{3}\); a negative input puts the angle in Quadrant II, \(\pi – \frac{\pi}{3} = \frac{2\pi}{3}\).
For arctan: tangent equals 1 at \(\frac{\pi}{4}\), so the negative case is \(-\frac{\pi}{4}\). The values are \(-\frac{\pi}{3}\), \(\frac{2\pi}{3}\) and \(-\frac{\pi}{4}\).
Try it
Question 1
Evaluate \(\cos^{-1}(0)\).
Show the answer
Cosine is 0 at \(\frac{\pi}{2}\), which lies in \([0, \pi]\), so the answer is \(\frac{\pi}{2}\).
Question 2
Why does \(\sin^{-1}\left(\sin\frac{5\pi}{6}\right)\) equal \(\frac{\pi}{6}\) rather than \(\frac{5\pi}{6}\)?
Show the answer
\(\sin\frac{5\pi}{6} = \frac{1}{2}\), and arcsin must return an angle in \([-\frac{\pi}{2}, \frac{\pi}{2}]\), where the angle with sine \(\frac{1}{2}\) is \(\frac{\pi}{6}\).
Inverse Trigonometric Functions vs. SOHCAHTOA
The right-triangle ratios take an angle and return a side ratio. These inverse functions run the opposite way: they take a ratio and return an angle. Also, \(\sin^{-1}x\) is not \(\frac{1}{\sin x}\); the reciprocal is the cosecant, written \(\csc x\).
Keep going
Lessons
Worksheets and quizzes
- Inverse Trigonometric Functions and Equations – Worksheet
- Inverse Trigonometric Functions and Equations – Quiz
Related words
- Unit Circle
- Trigonometric Identities
- Inverse Function
- One-to-One Function
- SOHCAHTOA
- Domain and Range
Whole dictionary: A–Z · Precalculus · Trigonometry