What do the reciprocal trig functions look like
Graphs of Inverse Trig Functions Name:
1) Let’s remember what happens when we graph the inverse of a function.
a) Find the inverse of the function [pic]
[pic]
On your graphing calculator, graph both [pic] and [pic]. Look at the table of values and the graphs to answer the questions below.
b) What happens graphically when you find the inverse of a function?
c) What happens to the domain and the range (or what happens to the x and the y) when you find the inverse of a function?
2) These same things happen when we graph the inverse of a trig function. Try to estimate and sketch (without using your graphing calculator) what the inverse of the sine function below would look like.
We can refer to the inverse of [pic] either as [pic]or as[pic].
What are the inputs and outputs of [pic]? Is [pic]a function? Explain why or why not.
Often in math, we will restrict the domain or the range of a relation so that something that is not a function becomes a function. This is frequently done with the inverse of sine, cosine, and tangent. Sometimes this is denoted by a capital A in Arcsin, Arccos, or Arctan and sometimes we just refer to the principal values of arcsine, arccosine, or arctangent.
The good news is that since the y= feature of the graphing calculator only graphs functions (and not relations that aren’t functions), the graphing calculator itself restricts the domain for us!
3) Let’s look at what these look like.
[pic] [pic] [pic]
4) Now let’s look at this inverse notation in another context. Of course, we use arcsine, arccosine, and arctangent to solve equations and find [pic]. But sometimes they show up in expressions.
Evaluate the expressions below. Feel free to use your calculator! If you don’t get an exact value, use your calculator for the inverse part and then draw a picture to finish the problem.
Find the exact value of each expression below. (When you’re asked for the principal value, that’s the value that is given by the calculator.)
|a) Find the principal value of [pic] |b) Find the principal value of [pic] |
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|c) Find [pic] |d) Find the principal value of [pic] |
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|e) Find [pic][pic] |f) Challenge:Express [pic] in terms of b. |
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[pic]
y = x
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