How do you graph inverse trigonometric functions?

Question

Oliver Atkinson · Accepted Answer

Since the graphs of #f(x)# and #f'(x)# are symmetric about the line #y=x#, start with the graph of a trigonometric function with an appropriate restricted domain, then reflect it about the line #y=x#.

(Caution: Their domains must be restricted to an appropriate interval so that their inverses exist.)

Let us sketch the graph of #y=sin^{-1}x#.

The graph of #y=sinx# on #[-pi/2, pi/2]# looks like:

By reflecting the graph above about the line #y=x#,

The curve in purple is the graph of #y=sin^{-1}x#.

The graphs of other inverse trigonometric functions can be obtained similarly.

I hope that this was helpful.

Kai Anderson · Answer

undefined To graph inverse trigonometric functions, you typically follow these steps:

1. **Understand the Function**: Have a clear understanding of the inverse trigonometric function you're dealing with. Common ones include arcsin (inverse sine), arccos (inverse cosine), and arctan (inverse tangent).

2. **Determine the Domain and Range**: Each inverse trigonometric function has a restricted domain and range to ensure it is one-to-one and has an inverse. For example, arcsin has a domain of [-1, 1] and a range of [-π/2, π/2], arccos has a domain of [-1, 1] and a range of [0, π], and arctan has a domain of all real numbers and a range of [-π/2, π/2].

3. **Plot Key Points**: Identify key points on the graph based on the function's domain and range. These points typically include the endpoints of the domain and the corresponding values of the function.

4. **Draw the Graph**: Connect the key points with a smooth curve, maintaining the characteristics of the function, such as concavity and direction.

5. **Label Axes and Units**: Label the x-axis and y-axis appropriately, indicating the units if necessary.

6. **Add Additional Information**: Depending on the context or specific requirements, you may need to add additional information to the graph, such as asymptotes or important intervals.

7. **Check Symmetry**: Inverse trigonometric functions often exhibit symmetry properties. For example, the arcsin function is symmetric about the line y = x, while arccos is symmetric about the line y = π - x.

By following these steps, you can accurately graph inverse trigonometric functions, providing visual representations of their behavior and characteristics.