Unit 5 · Topic 3.9
Inverse Trigonometric Functions
Use restricted trigonometric domains to turn ratios into unique principal angles and interpret inverse-function compositions.
Learning Goals
- Explain why periodic trigonometric functions need restricted domains before they can be inverted.
- State the domains and ranges of arcsine, arccosine, and arctangent.
- Evaluate exact and approximate inverse trigonometric values.
- Interpret graphs as reflections across \(y=x\).
- Evaluate compositions while respecting principal-value ranges.
1. Inverse Does Not Mean Reciprocal
The notation \(\sin^{-1}x\) means the inverse sine function, also written \(\arcsin x\). It returns an angle whose sine is \(x\).
Likewise, \(\cos^{-1}x\) is arccosine rather than secant, and \(\tan^{-1}x\) is arctangent rather than cotangent.
2. Why Restrict the Original Domain?
Sine, cosine, and tangent repeat outputs, so none is one-to-one over its natural domain. Their complete graphs fail the horizontal-line test.
Restricting each original function to one carefully chosen interval preserves its full range while making every output correspond to exactly one input. Only then does its inverse pass the vertical-line test.
3. Restricted Domains and Principal Ranges
| Original function | Restricted domain | Inverse function | Inverse domain | Principal range |
|---|---|---|---|---|
| \(\sin x\) | \([-\pi/2,\pi/2]\) | \(\arcsin x\) | \([-1,1]\) | \([-\pi/2,\pi/2]\) |
| \(\cos x\) | \([0,\pi]\) | \(\arccos x\) | \([-1,1]\) | \([0,\pi]\) |
| \(\tan x\) | \((-\pi/2,\pi/2)\) | \(\arctan x\) | \((-\infty,\infty)\) | \((-\pi/2,\pi/2)\) |
The principal range is the only interval from which that inverse function may return an angle.
4. Read Each Definition in Both Directions
An inverse-trigonometric input is a trigonometric ratio; its output is a principal angle, usually measured in radians unless a context specifies degrees.
5. Reflect the Restricted Graph
Interchanging every point \((a,b)\) with \((b,a)\) reflects a restricted trigonometric graph across \(y=x\). The original domain becomes the inverse range, and the original range becomes the inverse domain.
6. Compare the Three Inverse Graphs
- \(\arcsin x\) is increasing from \((-1,-\pi/2)\) to \((1,\pi/2)\).
- \(\arccos x\) is decreasing from \((-1,\pi)\) to \((1,0)\).
- \(\arctan x\) is increasing for all real inputs and approaches horizontal asymptotes \(y=\pm\pi/2\).
Arcsine and arccosine include their endpoint outputs. Arctangent never reaches either horizontal asymptote.
7. Exact Principal Values
| Expression | Principal angle | Reason |
|---|---|---|
| \(\arcsin(1/2)\) | \(\pi/6\) | \(\sin(\pi/6)=1/2\) |
| \(\arcsin(-\sqrt3/2)\) | \(-\pi/3\) | Output must lie in \([-\pi/2,\pi/2]\) |
| \(\arccos(-\sqrt2/2)\) | \(3\pi/4\) | Output must lie in \([0,\pi]\) |
| \(\arctan(-1)\) | \(-\pi/4\) | Output must lie in \((-\pi/2,\pi/2)\) |
The unit circle may contain several angles with the requested ratio, but the principal range selects exactly one.
8. A Principal-Value Workflow
- Identify whether the inverse is arcsine, arccosine, or arctangent.
- Write its permitted output interval.
- Find a reference angle with the requested ratio.
- Choose the angle in the permitted interval with the correct sign.
- Substitute into the original trigonometric function to verify.
9. Use Technology with the Correct Mode
For a non-special value, technology gives an approximation. In radian mode,
In degree mode, the same angle is approximately \(113.58^\circ\). State the unit and requested precision; a correct key sequence in the wrong angle mode gives a numerically different display.
10. Original after Inverse
When the inverse function is applied first, the original function undoes it throughout the inverse domain:
The stated domains matter. For example, \(\arcsin(2)\) has no real value.
11. Inverse after Original Needs More Care
The reverse composition returns the principal angle equivalent to the original input, not necessarily the original input itself.
| Expression | Value | Why it changes |
|---|---|---|
| \(\arcsin(\sin(3\pi/4))\) | \(\pi/4\) | \(3\pi/4\) is outside the arcsine range |
| \(\arccos(\cos(5\pi/3))\) | \(\pi/3\) | Arccosine returns an angle in \([0,\pi]\) |
| \(\arctan(\tan(3\pi/4))\) | \(-\pi/4\) | Arctangent returns an angle in \((-\pi/2,\pi/2)\) |
The identity \(f^{-1}(f(x))=x\) is guaranteed only when \(x\) lies in the restricted domain chosen for \(f\).
12. Evaluate a Mixed Composition
Find \(\sin(\arctan(3/4))\). Let \(\theta=\arctan(3/4)\), so \(\tan\theta=3/4\). Since \(\theta\in(-\pi/2,\pi/2)\) and the ratio is positive, \(\theta\) is in Quadrant I.
Use a right triangle with opposite side 3 and adjacent side 4. Its hypotenuse is 5, so
The principal range determines the quadrant and therefore the signs of all triangle ratios.
13. Derive an Algebraic Composition
Let \(\theta=\arccos x\). Then \(\cos\theta=x\) and \(0\le\theta\le\pi\), where sine is nonnegative. From the Pythagorean identity,
The nonnegative square root is required by the principal range of arccosine.
14. Apply an Inverse Ratio in Context
A ramp rises 7 units over a horizontal run of 24 units. Its angle of elevation \(\theta\) satisfies
The positive acute principal value matches the geometry. Include the angle unit and avoid rounding until the final step.
15. Common Errors
- Interpreting \(\sin^{-1}x\) as \(1/\sin x\).
- Returning every coterminal angle instead of one principal value.
- Using an arcsine input or arccosine input outside \([-1,1]\).
- Assuming \(\arcsin(\sin x)=x\) for every real \(x\).
- Choosing a reference angle outside the inverse function's range.
- Ignoring signs when building a triangle for a mixed composition.
- Mixing degree and radian modes or omitting units from an approximation.
Key Takeaways
- Use restricted domains and principal ranges to interpret inverse trigonometric values.
- Core relationship: \(\arcsin x=y\iff\sin y=x,\quad y\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\)
- Error check: Return the principal angle, not every coterminal solution.
- State the restricted domains used to define inverse sine, inverse cosine, and inverse tangent.
- Evaluate \(\arcsin(-1/2)\), \(\arccos(-1/2)\), and \(\arctan(\sqrt3)\).
- Explain why \(\arccos(\cos(7\pi/6))=5\pi/6\), not \(7\pi/6\).
- Find an exact expression for \(\cos(\arctan(5/12))\).
- A right triangle has opposite side 9 and adjacent side 14. Find its acute angle to the nearest tenth of a degree.