8.2 Inverse Trigonometric Functions
Algebra Refresher
Review the following skills you will need for this section.
Find a formula for the inverse function.
State the domain and range of the inverse function.
Graph the function and its inverse on the same grid.
[latex]\displaystyle f(x) = \dfrac{1}{2}x-4[/latex]
[latex]\displaystyle g(x) = 3x+6[/latex]
[latex]\displaystyle F(x) = 2+\dfrac{1}{x}[/latex]
[latex]\displaystyle G(x) = \dfrac{1}{x+5}[/latex]
[latex]\displaystyle h(x) = \sqrt{x+2}[/latex]
[latex]\displaystyle H(x) = 3+\sqrt[3]{x}[/latex]
Algebra Refresher Answers
[latex]\displaystyle f^{-1}(x)=2x+8[/latex]
Dom: [latex](-\infty, \infty)[/latex] Rge: [latex](-\infty, \infty)[/latex]
[latex]\displaystyle g^{-1}(x)=\dfrac{1}{3}x-2[/latex]
Dom: [latex](-\infty, \infty)[/latex] Rge: [latex](-\infty, \infty)[/latex]
[latex]\displaystyle F^{-1}(x)=\dfrac{1}{x-2}[/latex]
Dom: [latex]x \not=0[/latex] Rge: [latex]y \not=2[/latex]
[latex]\displaystyle G^{-1}(x)=\dfrac{1}{x}-5[/latex]
Dom: [latex]x \not=-5[/latex] Rge: [latex]y \not=0[/latex]
[latex]\displaystyle h^{-1}(x)=x^2-2[/latex]
Dom: [latex]x \ge 0[/latex] Rge: [latex]y \ge -2[/latex]
[latex]\displaystyle H^{-1}(x)=(x-3)^3[/latex]
Dom: [latex](-\infty, \infty)[/latex] Rge: [latex](-\infty, \infty)[/latex]
- Decide whether a function has an inverse function
- Evaluate the inverse trig functions
- Model problems with inverse trig functions
- Solve formulas
- Simplify expressions involving the inverse trig functions
- Graph the inverse trig functions
We have been using the calculator keys [latex]\boxed{SIN^{-1}}{,}[/latex] [latex]\boxed{COS^{-1}}{,}[/latex] and [latex]\boxed{TAN^{-1}}[/latex] to find approximate values of [latex]\theta[/latex] when we know either [latex]\sin \theta,~ \cos \theta{,}[/latex] or [latex]\tan \theta{.}[/latex] For example, if we know that [latex]\cos \theta = 0.3{,}[/latex] then
[latex]\theta = \cos^{-1}(0.3) \approx 1.2661~ {radians}[/latex]
In other words, we use the [latex]\boxed{SIN^{-1}}{,}[/latex] [latex]\boxed{COS^{-1}}{,}[/latex] and [latex]\boxed{TAN^{-1}}[/latex] keys to solve trigonometric equations, just as we use square roots to solve quadratic equations. Using one of these keys performs the inverse operation for computing a sine, cosine or tangent, just as extracting square roots is the inverse of squaring a number. Many functions can be described as an operation or as a sequence of operations on the input value, and this leads us to the notion of an inverse function.
Raising a number to the [latex]n^{th}[/latex] power and taking [latex]n^{th}[/latex] roots are an example of inverse operations. For example, if we first cube a number and then take the cube root of the result, we return to the original number.
We say that the two functions [latex]f(x)=x^3[/latex] and [latex]g(x)=\sqrt[3]{x}[/latex] are inverse functions. Each of the functions undoes the results of the other function. You can confirm this behavior by consulting the tables of values for the