This section covers:

**Introduction to Inverse Trig Functions****Graphs of Inverse Functions****Evaluating Inverse Trig Functions – Special Angles****Transformations of the Inverse Trig Functions****Composite Inverse Trig Functions with Non-Special Angles****More Practice**

# Introduction to Inverse Trig Functions

We studied **Inverses of Functions** here; we remember that getting the inverse of a function is basically switching the \(x\) and \(y\) values, and the inverse of a function is symmetrical (a mirror image) around the line \(y=x\).

The same principles apply for the **inverses of six trigonometric functions**, but since the trig functions are periodic (repeating), these functions don’t have inverses, unless we **restrict the domain**. As shown below, we will restrict the domains to certain **quadrants** so the original function passes the **horizontal line test** and thus the inverse function passes the **vertical line test**.

Note that if \({{\sin }^{-1}}\left( x \right)=y\), then \(\sin \left( y \right)=x\). When we take the inverse of a trig function, what’s in parentheses (the \(x\) here), is **not an angle**, but the actual **sin** (trig) value. The trig inverse (the \(y\) above) is the angle (usually in radians).

Also note that the **–1** is **not an exponent**, so we are not putting anything in a denominator.

We can also write trig functions with “arcsin” instead of \({{\sin }^{-1}}\): if \(\arcsin \left( x \right)=y\), then \(\sin \left( y \right)=x\).

Let’s show how quadrants are important when getting the inverse of a trig function using the **sin** function. In order to make an inverse trig function an actual function, we’ll only take the values between \(\displaystyle -\frac{\pi }{2}\) and \(\displaystyle \frac{\pi }{2}\), so the sin function passes the horizontal line test (meaning its inverse is a function):

To help remember which quadrants the inverse trig functions will come from, I use these “sun” diagrams:

The inverse **cos**, **sec**, and **cot** functions will return values in the **I and II Quadrants**, and the inverse **sin**,** csc**,** and tan** functions will return values in the **I and IV Quadrants **(but remember that you need the **negative values** in** Quadrant IV**). (I would just memorize these, since it’s simple to do so). These are called **domain restrictions** for the inverse trig functions.

**Important Note: **There is a subtle distinction between **finding inverse trig functions** and **solving for trig functions**. If we want \(\displaystyle {{\sin }^{{-1}}}\left( {\frac{{\sqrt{2}}}{2}} \right)\) for example, we only pick the answers from **Quadrants** **I** and **IV**, so we get \(\displaystyle \frac{\pi }{4}\) only. But if we are solving \(\displaystyle \sin \left( x \right)=\frac{{\sqrt{2}}}{2}\) like in the **Solving Trigonometric Functions** section, we get \(\displaystyle \frac{\pi }{4}\) and \(\displaystyle \frac{{3\pi }}{4}\) in the interval \(\left( {0,2\pi } \right)\); there are no domain restrictions.

# Graphs of Inverse Trig Functions

Here are tables of the inverse trig functions and their **t-charts**, **graphs**, **domain **and** range **(also called the** principal interval**). First, **inverse sin** and **inverse cos**:

Here are the **inverse tan** and **cot** functions. Notice that the **tan** and **cot** inverse functions come from different sets of quadrants: **tan** from Quadrants I and IV, and **cot** from Quadrants I and II:

And here are the **inverse csc** and **sec** functions:

# Evaluating Inverse Trig Functions – Special Angles

When you are asked to evaluate inverse functions, you may be see the notation like \({{\sin }^{-1}}\) or **arcsin**.

The following examples makes use of the fact that the angles we are evaluating are **special values or special angles**, or angles that have trig values that we can compute exactly (they come right off the **Unit Circle** that we have studied).

Here is the **Unit Circle** again:

To do these problems, use the **Unit Circle** remember again the “sun” diagrams to make sure you’re getting the angle back from the correct quadrant:

When using the **Unit Circle**, when the answer is in **Quadrant IV**, it must be negative (go backwards from the \((1, 0)\) point). For example, for the \(\displaystyle {{\sin }^{-1}}\left( -\frac{1}{2} \right)\) or \(\displaystyle \arcsin \left( -\frac{1}{2} \right)\), we see that the angle is **330°**, or \(\displaystyle \frac{11\pi }{6}\). But since our answer has to be between \(\displaystyle -\frac{\pi }{2}\) and \(\displaystyle \frac{\pi }{2}\), we need to change this to the co-terminal angle \(-30{}^\circ \), or \(\displaystyle -\frac{\pi }{3}\).

To get the inverses for the **reciprocal functions**, you do the same thing, but we’ll take the reciprocal of what’s in the parentheses and then use the “normal” trig functions. For example, to get \({{\sec }^{-1}}\left( -\sqrt{2} \right)\), we have to look for \(\displaystyle {{\cos }^{-1}}\left( -\frac{1}{\sqrt{2}} \right)\), which is \(\displaystyle {{\cos }^{-1}}\left( -\frac{\sqrt{2}}{2} \right)\), which is \(\displaystyle \frac{3\pi }{4}\), or **135°**.

## Trig Inverses in the Calculator

You can also put trig inverses in the graphing calculator and use the **2**^{nd} button before the trig functions: ; however, with radians, you won’t get the exact answers with \(\pi \) in it. (In the **degrees** mode, you will get the degrees.) Here’s an example in **radian mode**: , and in **degree mode**: .

For the **reciprocal functions** (**csc**, **sec**, and **cot**), you take the reciprocal of what’s in parentheses, and then use the “normal” trig functions in the calculator. For example, to put \({{\sec }^{-1}}\left( -\sqrt{2} \right)\) in the calculator (degrees mode), you’ll use \({{\cos }^{-1}}\) as follows: .

When you are getting the **arccot** or \({{\cot }^{-1}}\) of a **negative number**, you have to add \(\pi \) to the answer that you get (or **180°** if in degrees); this is because **arccot** come from Quadrants I and II, and since we’re using the **arctan** function in the calculator, we need to add \(\pi \). Here is example of getting \(\displaystyle {{\cot }^{-1}}\left( -\frac{1}{\sqrt{3}} \right)\) in radians: , or in degrees: .

**Note**: For all inverse trig functions of a positive argument (other than **1**), we should get an angle in Quadrant I (\(\displaystyle 0\le \theta \le \frac{\pi }{2}\)). For the arcsin, arccsc, and arctan functions, if we have a negative argument, we’ll end up in Quadrant IV (specifically \(\displaystyle -\frac{\pi }{2}\le \theta \le \frac{\pi }{2}\)), and for the arccos, arcsec, and arccot functions, if we have a negative argument, we’ll end up in Quadrant II (\(\displaystyle \frac{\pi }{2}\le \theta \le \pi \)).

Here are more problems:

# Transformations of the Inverse Trig Functions

We learned how to transform Basic Parent Functions here in the **Parent Functions and Transformations** section, and we learned how to transform the **six Trigonometric Functions**** here.**

Now we will transform the **Inverse Trig Functions**.

## T-Charts for the Six Inverse Trigonometric Functions

Some prefer to do all the transformations **with** **t-charts** like we did earlier, and some prefer it **without t-charts**; most of the examples will show t-charts.

Here are the **inverse** **trig parent function t-charts** I like to use. Note that each is in the **correct quadrants** (in order to make true functions).

Note also that when the original functions have **0**’s as \(y\) values, their respective reciprocal functions are **undefined** (undef) at those points (because of division of **0**); these are **vertical asymptotes**.

And remember that arcsin and \({{\sin }^{-1}}\) , for example, are the same thing.

Here are examples, using t-charts to perform the transformations. Remember that when functions are transformed on the **outside** of the function, or parentheses, you move the function up and down and do the “**regular**” math, and when transformations are made on the** inside** of the function, or parentheses, you move the function **back and forth**, but do the “opposite math”:

Here are examples of reciprocal trig function transformations:

# Composite Inverse Trig Functions with Special Values/Angles

Sometimes you’ll have to take the trig function of an inverse trig function; sort of “undoing” what you’ve just done (called composite inverse trig functions).

We still have to remember which quadrants the inverse (inside) trig functions come from:

Let’s start with some examples with the **special values **or** special angles**, meaning the “answers” will be on the unit circle:

## Trig Composites on the Calculator

You can also put trig composites in the graphing calculator (and they don’t have to be special angles), but remember to add \(\pi \) to the answer that you get (or **180°** if in degrees) when you are getting the **arccot** or \({{\cot }^{{-1}}}\) of a **negative number** (see last example). (I checked answers for the exact angle solutions).

Note again for the reciprocal functions, you put **1** over the whole trig function when you work with the regular trig functions (like **cos**), and you take the reciprocal of what’s in the parentheses when you work with the inverse trig functions (like **arccos**).

Some examples:

# Composite Inverse Trig Functions with Non-Special Angles

You will also have to find the composite inverse trig functions with **non-special angles**, which means that they are not found on the Unit Circle. Examples of **special angles** are **0°, 45°, 60°, 270°,** and their radian equivalents.

The easiest way to do this is to draw triangles on they coordinate system, and (if necessary) use the **Pythagorean Theorem** to find the missing sides.

Remember that the \(r\) (hypotenuse) **can never be negative**!

To know where to put the triangles, use the “**bowtie**” hint: always make the triangle you draw as part of a bowtie that sits on the \(x\)-axis. Note that the triangle needs to “hug” the \(x\)-axis, not the \(y\)-axis:

We find the values of the composite trig functions (inside) by drawing triangles, using **SOH-CAH-TOA**, or the **trig definitions** found here in the **Right Triangle Trigonometry** Section, and then using the **Pythagorean Theorem **to determine the unknown sides. Then we use SOH-CAH-TOA again to find the (outside) trig values. We still have to remember which quadrants the inverse (inside) trig functions come from:

**Note**: If the angle we’re dealing with is on one of the axes, such as with the arctan(**0°**), we don’t have to draw a triangle, but just draw a line on the \(x\)** **or

**\(y\)-axis.**

Let’s do some problems. Remember again that \(r\) (hypotenuse of triangle) is never negative, and when you see whole numbers as arguments, use **1** as the denominator for the triangle. Also note that you’ll **never be drawing a triangle in Quadrant III** for these problems.

Here are some problems where we have **variables** in the side measurements. Note that the algebraic expressions are still based on the Pythagorean Theorem for the triangles, and that \(r\) (hypotenuse) is never negative.

Assume that all variables are positive, and note that I used the variable \(t\) instead of \(x\) to avoid confusion with the \(x\)’s in the triangle:

**Learn these rules, and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

On to **Solving Trigonometric Equations **– you are ready!