How to Graph Inverse of the Sine Function?
Tutor-style math help
Graph Inverse of the Sine Function: what to notice and how to work it
Trigonometry skill
Inverse trig graphs reverse a restricted trig function. The key is knowing the allowed domain and range for the inverse relation.
What to notice first
Read inverse trig graphs by domain and range first. For example, \(y=\sin^{-1}x\) uses inputs from -1 to 1 and outputs from \(-\pi/2\) to \(\pi/2\).
Common student mistake
Do not graph the full sine, cosine, or tangent curve backward. The original trig function must be restricted before it has an inverse function.
Key formulas and cues
\(y=\sin^{-1}x:\ -1\le x\le1,\ -\frac{\pi}{2}\le y\le\frac{\pi}{2}\)
\(y=\cos^{-1}x:\ -1\le x\le1,\ 0\le y\le\pi\)
\(y=\tan^{-1}x:\ -\frac{\pi}{2}<y<\frac{\pi}{2}\)
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Read arcsine
Example: Find the domain and range of \(y=\sin^{-1}x\).
- Arcsine reverses the restricted sine graph.
- The input must be a sine output, so x is between -1 and 1.
- The principal outputs run from -pi/2 to pi/2.
Answer: Domain \([-1,1]\), range \([-\pi/2,\pi/2]\).
Read arctangent
Example: What are the horizontal asymptotes of \(y=\tan^{-1}x\)?
- Arctangent's outputs approach pi/2 and -pi/2.
- Those are y-values, so they are horizontal asymptotes.
- The graph never reaches them.
Answer: \(y=\pm\frac{\pi}{2}\).
Try one before moving on
Try: Give the range of \(y=\tan^{-1}x\).
Answer: \((-\pi/2,\pi/2)\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Graph Inverse of the Sine Function: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. The domain of \(y=\sin^{-1}x\) is:
2. The range of \(y=\cos^{-1}x\) is:
3. Inverse trig graphs come from:
A step-by-step guide to graphing the inverse of the sine function
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