The Inverse Trigonometric Functions

This section covers:

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.  So, 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.

So 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   \(-\frac{\pi }{2}\)  and  \(\frac{\pi }{2}\), so the sin function passes the horizontal line test (meaning its inverse is a function):

Graph of Inverse Sin Function

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

Inverse Trig SunsSo 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  \({{\sin }^{{-1}}}\left( {\frac{{\sqrt{2}}}{2}} \right)\)  for example, we only pick the answers from Quadrants I and IV, so we get  \(\frac{\pi }{4}\) only.  But if we are solving  \(\sin \left( x \right)=\frac{{\sqrt{2}}}{2}\)  like in the Solving Trigonometric Functions section, we get  \(\frac{\pi }{4}\)  and  \(\frac{{3\pi }}{4}\)  in the interval (0, 2π); 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:

Inverse Sin and Cos Graphs

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:

Inverse Tan and Cot Graphs

And here are the inverse csc and sec functions:

Inverse Csc and Sec Graphs

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:

Unit Circle

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:

Inverse Trig Suns

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  \({{\sin }^{-1}}\left( -\frac{1}{2} \right)\)  or  \(\arcsin \left( -\frac{1}{2} \right)\),   we see that the angle is 330°, or  \(\frac{11\pi }{6}\).  But since our answer has to be between   \(-\frac{\pi }{2}\)  and    \(\frac{\pi }{2}\), we need to change this to the co-terminal angle  \(-30{}^\circ \), or  \(-\frac{\pi }{3}\).

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  \({{\cos }^{-1}}\left( -\frac{1}{\sqrt{2}} \right)\), which is  \({{\cos }^{-1}}\left( -\frac{\sqrt{2}}{2} \right)\), which is  \(\frac{3\pi }{4}\), or 135°.

Trig Inverses in the Calculator

You can also put trig inverses in the graphing calculator and use the 2nd button before the trig functions:  Inverse Cos in Calculator; however, with radians,  you won’t get the exact answers with  π in it.  (In the degrees mode, you will get the degrees.)    Here’s an example in radian mode:   cos -1 in calc, and in degree mode:  cos -1 in calc degrees.

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:  Cot Example in Calculator.    Now, when you are getting the arccot or  \({{\cot }^{-1}}\) of a negative number, you have to add π 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 π.  Here is example of getting  \({{\cot }^{-1}}\left( -\frac{1}{\sqrt{3}} \right)\)  in radians:  Inverse Cot in Calc, or in degrees:  Inverse Cot in Calc Degrees.

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

Here are more problems:

Inverse Trig 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.

Inverse Trig Function T Charts

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”:

Transforming Inverse Trig Functions

Here are examples of reciprocal trig function transformations:

Transforming Inverse Reciprocal Trig Functions

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:

Inverse Trig Suns

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

Composite Inverse Trig Functions Special Angles

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 π 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:

Trig Composite Calculator

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:

Trig Bow Tie

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:

Inverse Trig Suns

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.

Composite Inverse Trig Functions Non-Special Angles

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:

Composite Inverse Trig With Variables

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!

6 thoughts on “The Inverse Trigonometric Functions

  1. Why do use radians and degrees when the x axis consisting of the real numbers is being wrapped around the unit circle. The domain of sin x is all reals… not degrees…………technically you compromise the whole concept of the Wrapping Function …

    • Thank you so so much! I can’t believe I didn’t catch this – I must be staring at this stuff too long 😉 PLEASE let me know if you see anything else. Lisa 🙂

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