*Yuanxin (Amy) Yang Alcocer*Show bio

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*
Show bio

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will understand why our inverse trigonometric functions have a limited range. Learn what this does. Also learn the notation for these functions.

In addition to the trigonometric functions that we are familiar with at this point such as sine, cosine, and tangent, we also have what are called the **inverse trigonometric functions**. It is these functions that we will be talking about in this video lesson. What exactly are they? They are the inverse functions of our trig functions.

However, in trigonometry, the inverse function here is not 1 divided by the function. This inverse function allows you to solve for the argument. For example, if you have the problem *sin x = 1*, we can solve the problem by multiplying both sides by the inverse sine function. The inverse sine function cancels the sine function on the left side and we are left with *x = sin^-1 (1)*. Evaluating the right side allows us to find the angle of the sine function that fits the problem.

As you have just seen, the notation for these inverse trigonometric functions is unique. We use an exponent of -1 to let us know that we are dealing with the inverse trig function. We can write our inverse trig functions like this:

This is the notation that you will see most often in textbooks and various trig problems. And this is most likely the notation you will use when writing out your problems. But, in trigonometry, we also have formal names for these functions. We call the inverse sine function the arcsine function, the inverse cosine function the arccosine function, and the inverse tangent function the arctangent function. While you will see the first notation more often in problems, you will come across these formal names in math discussions. So, it's good to know both. It's easy to remember these names if you link the arc with the -1 exponent. All the inverse trigonometric functions begin with the prefix *arc-* followed by the name of the trig function that we already know.

Now, think back to the regular trig functions of sine, cosine, and tangent. Do you remember their graphs?

The red line is the graph of the *sine* function, the blue line the graph of the *cosine* function and the purple line is the graph of the *tangent* function. What interesting thing do you notice about these graphs?

All of these graphs repeat every so often. The tangent function repeats every pi spaces while the sine and cosine functions repeat every 2pi spaces. Each time the function repeats, we get the same output answer. Because these functions repeat, we have to limit the range, or output values, of our inverse trig functions. Otherwise, we would get different answers each time.

By limiting the range of our inverse function, we find the principal or primary value of our inverse function. This is what is going inside your calculator whenever you perform an inverse trig function. It gives you an answer within the accepted range. If we didn't limit our range, our calculator wouldn't know which answer to give you since the answers repeat every 2pi for the sine and cosine functions and pi for the tangent function. The following are the accepted limited ranges for our inverse trigonometric functions:

Inverse function | Range |
---|---|

y = arcsin x | -pi/2 <= y <= pi/2 |

y = arccos x | 0 <= y <= pi |

y = arctan x | -pi/2 < y < pi/2 |

These ranges don't exactly correspond to how our regular trig functions repeat. This time, we have the inverse cosine function that is limited between 0 and pi. The inverse sine function is limited to between -pi/2 and pi/2 including those points. The inverse tangent function has the same limited range as the inverse sine except the two points of -pi/2 and pi/2 are not included.

What do these limits mean? These limits tell you that the answer you will get from your calculator will be within those limits. And if you are calculating by hand, these limits tell you that your principal answer needs to be within this range, too. But keep in mind that the answer you get is just the primary answer. You have more answers spaced every 2pi numbers apart for inverse cosine and sine, and every pi numbers apart for inverse tangent.

For example, *sin^-1 (1) = pi/2*. This is the primary answer, but in reality we have answers every 2pi apart. We can include all our answers by writing *pi/2 + 2*pi*n* where *n* is the number of spaces the answer is away from the primary answer.

Because our inverse functions are limited to their range, so is our function when we graph it. Instead of our functions continuing forever like our sine, cosine, and tangent graphs, our arcsine, arccosine, and arctangent graphs only show the graph within the accepted limited range.

This graph shows the *arcsine* function as the red line, the *arccosine* function as the blue line, and the *arctangent* function as the purple line:

See how we have limited the graph of each of these functions? If we didn't limit them, these graphs would continue forever repeating themselves over and over just like our sine, cosine, and tangent functions.

Let's review what we've learned. **Inverse trigonometric functions** are the inverse functions of our trig functions. Our trig functions are our usual functions of sine, cosine, and tangent. There are two ways to write our inverse functions. We can call them by name. We have the inverse of sine is arcsine, the inverse of cosine is arccosine, and the inverse of tangent is arctangent. We can also write them using the -1 exponent symbol.

This inverse function allows us to find the angle of a trig function. For example, to find the angle for the problem *sin x = 1*, we apply the inverse sine function to both sides of the equation. It cancels with the sine function on the left side and we are left with *x = sin^-1 (1)*. We evaluate the right side to find our answer. We can use our calculator to find the answer. If we do, we will get the primary answer.

Remember, our sine and cosine functions repeat every 2pi spaces and our tangent function repeats every pi spaces. Because our inverse functions are limited to the primary answer, each inverse function also has a limited range. They are as follows:

Inverse function | Range |
---|---|

y = arcsin x | -pi/2 <= y <= pi/2 |

y = arccos x | 0 <= y <= pi |

y = arctan x | -pi/2 < y < pi/2 |

The graphs of the inverse functions also shows this limited range. This graph shows the *arcsine* function as the red line, the *arccosine* function as the blue line, and the *arctangent* function as the purple line:

Once you've finished with this lesson, you will have the ability to:

- Identify the inverse trigonometric functions and their graphs
- Describe the two ways to write these functions
- Explain the limited range of inverse functions

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