Precalculus Inverse Trigonometric Functions and Equations Lesson (Free PDF)
This inverse trig lesson for Precalculus covers inverse trigonometric functions and equations (Common Core F-TF.B.6, F-TF.B.7), with a free PDF to print.
Inverse trig functions return angles: arcsin x is in [-pi/2, pi/2], arccos x is in [0, pi] and arctan x is in (-pi/2, pi/2). To solve a trig equation, isolate the function, find the reference angle, list every solution in the interval using the quadrants, and add the period (2 pi k or pi k) for all solutions.
Precalculus · Common Core State Standards: F-TF.B.6, F-TF.B.7
Key facts in this inverse trig lesson
- arcsin and arctan have range (or closed range) centered on 0; arccos has range [0, pi].
- sin(arcsin x) = x on [-1, 1], but arcsin(sin x) = x only for x in [-pi/2, pi/2].
- sin^-1 x means the inverse function, not 1/sin x.
- Sine and cosine equations usually have two solutions per period; tangent has one.
- Factor quadratic-type trig equations before solving.
What are the inverse trigonometric functions?
Sine, cosine and tangent are not one-to-one, so each is restricted to an interval where it is. The inverse functions are written \(\sin^{-1}x\) or \(\arcsin x\), and so on. The notation \(\sin^{-1}x\) means the inverse function, not \(\frac{1}{\sin x}\).
| Function | Domain | Range |
|---|---|---|
| \(\arcsin x\) | \([-1,1]\) | \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) |
| \(\arccos x\) | \([-1,1]\) | \([0,\pi]\) |
| \(\arctan x\) | all real numbers | \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) |
To evaluate, ask: which angle in the range has this value? \(\arcsin\frac12=\frac{\pi}{6}\), \(\arccos\left(-\frac{\sqrt2}{2}\right)=\frac{3\pi}{4}\), and \(\arctan\sqrt3=\frac{\pi}{3}\).
How do you evaluate compositions?
\(\sin(\arcsin x)=x\) for \(x\) in \([-1,1]\). But \(\arcsin(\sin x)=x\) only when \(x\) is in \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). For example \(\arcsin\left(\sin\frac{7\pi}{6}\right)=-\frac{\pi}{6}\), because the answer must be in the range. For mixed compositions, draw a right triangle: if \(\theta=\arcsin\frac35\), then \(\sin\theta=\frac35\) and the third side is 4, so \(\cos\theta=\frac45\).
How do you solve a trigonometric equation?
- Isolate the trig function, for example \(\sin x=\frac12\).
- Find the reference angle with an inverse function or the unit circle.
- Use the quadrants to list all solutions in the interval.
- For all solutions, add the period: \(2\pi k\) for sine and cosine, \(\pi k\) for tangent.
If an equation is quadratic in a trig function, factor it first. For \(2\sin^{2}x-\sin x-1=0\), factor \((2\sin x+1)(\sin x-1)=0\), so \(\sin x=-\frac12\) or \(\sin x=1\), giving \(x=\frac{7\pi}{6},\frac{11\pi}{6},\frac{\pi}{2}\).
Inverse trig functions also turn side lengths into angles. In a right triangle with opposite side \(a\), adjacent side \(b\) and hypotenuse \(c\), the acute angle \(\theta\) satisfies \(\theta=\arcsin\frac{a}{c}=\arccos\frac{b}{c}=\arctan\frac{a}{b}\). A ramp that rises 2 feet over a 12-foot length makes an angle of \(\arcsin\frac{2}{12}\approx9.6^{\circ}\). This is how surveyors, builders and pilots find angles from measured distances.
When the value is not a special angle, use a calculator in radian mode. For \(\cos x=0.3\), \(x=\arccos0.3\approx1.266\) and the second solution is \(2\pi-1.266\approx5.017\).
Inverse trig lesson: questions and answers
Why are the ranges of inverse trig functions restricted?
A function needs one output for each input. Restricting sine, cosine and tangent to intervals where they are one-to-one gives each inverse exactly one answer.
Is sin^-1 x the same as 1/sin x?
No. The exponent -1 on a function name means inverse. The reciprocal 1/sin x is written (sin x)^-1 or csc x.
Why is arcsin(sin(7 pi/6)) not 7 pi/6?
The output of arcsin must lie in [-pi/2, pi/2]. Since sin(7 pi/6) = -1/2, arcsin(-1/2) = -pi/6, which is in that interval.
How many solutions does sin x = 1/2 have?
In one period [0, 2 pi) it has two: pi/6 and 5 pi/6. On all real numbers it has infinitely many, written pi/6 + 2 pi k and 5 pi/6 + 2 pi k.
Should the calculator be in degree or radian mode?
Match the problem. If angles are given in radians, use radian mode. A mode mismatch is a common cause of wrong answers.
How to use this inverse trig lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the inverse trig lesson as a PDF with the button below and keep it with your notes.
Download the inverse trig lesson as a PDF with the button below and keep it with your notes.
All Precalculus chapters: Precalculus math.
