# Inverse Trig Functions

Inverse trig functions are a core concept in trigonometry, and you’re probably already used to using them without even knowing what they are.  Specifically, they are named arcsin, arccos, and arctan (or, their full names: arcsine, arccosine, arctangent). We have already discussed how to find things such as the sine of 25 degrees, or cosine of 71 degrees.  However, we need to introduce a new math concept to figure out such problems as “what angle gives a sine value of 0.2?” This is where inverse trig functions come in. With these functions, you start with the trig ratio and finish with the angle. Think of their relationship to the standard basic trig functions as being comparable to that between multiplication and division.  You do one operation, and then the opposite to go the other way and undo the first operation. One way takes the sine of an angle to get the ratio. The other way takes the arcsine of the ratio to get the angle. Take a look at Math Open Reference to get another way of explaining it.

You may be taught these in school slightly differently.  Instead of using the names arcsine, arccosine, and arctangent (arcfunctions!), it is also very common to see these represented as the basic trig notation with a -1 exponent, like these:

arcsine = sin-1

arccosine = cos-1

arctangent = tan-1

(Interestingly, SparkNotes indicates that these functions are typically written with a capital letter, as in Arcsin, though I don’t seem to see that convention commonly used elsewhere. Leave me a comment below if you know more about this.)

These are likely the symbols that you will see on your calculator.  Typically, they are the shifted function of the regular sin, cos, and tan buttons.  It is important to note the distinction between the basic trigonometric functions and these new inverse trig functions.  When using the basic trig functions, the value you are obtaining is the ratio of the two relevant sides of the triangle for the given angle.  When using the inverse trig functions, what you are solving for is the actual angle that produces the given ratio of sides.  So, make sure you push the right button on the calculator! Also, it is important to realize that these -1 exponent notations are NOT actual exponents indicating to take 1/sine, etc. They are simply a notation to imply the inverse trig functions.

Now, follow along with an easy trick that I explain below to help you work with inverse trig functions, and then go check out Math Warehouse, where there is a really good explanation, as well as a bunch of problems and a video for working with inverse sine, cosine and tangent.

The most basic way of finding an inverse function, in general, is a trick I was taught long ago, where you take your given function, substitute y for f(x), switch the x and y, and then rearrange.

f(x) = x2 + 5

y = x2 + 5…. now switch x and y

x = y2 +5…. and rearrange

y2 = x – 5

y = sqrt(x-5)

And there you have determined the inverse function.  This is the same strategy that is being applied when we are talking about inverse trig functions.  However, having the inverse trig buttons on our calculators really take all of this extensive and possibly difficult rearranging and calculating out of the picture.

Here is a basic example of one of these inverse trigonometric functions.  Hopefully you will see that they are extremely easy to work with.

Find the angle for the given trig ratio:

sin(θ) = 1 /√2

θ = sin-1(1 /√2)

θ = 45°