We know that \(\sin (30^o)=\dfrac{1}{2}\). Now, suppose we are asked to find an angle,\(\theta\), whose sine is \(\dfrac{1}{2}\). We can't solve this problem with the normal trigonometric functions, we need inverse trigonometric functions! What are those?
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Jetzt kostenlos anmeldenWe know that \(\sin (30^o)=\dfrac{1}{2}\). Now, suppose we are asked to find an angle,\(\theta\), whose sine is \(\dfrac{1}{2}\). We can't solve this problem with the normal trigonometric functions, we need inverse trigonometric functions! What are those?
In this article, we go over what inverse trigonometric functions are and discuss their formulas, graphs, and examples in detail. But before moving on, if you need to review inverse functions, please refer to our Inverse Functions article.
From our Inverse Functions article, we remember that the inverse of a function can be found algebraically by switching the x- and y-values and then solving for y. We also remember that we can find the graph of the inverse of a function by reflecting the graph of the original function over the line \(y=x\).
We already know about inverse operations. For instance, addition and subtraction are inverses, and multiplication and division are inverses.
The key here is: an operation (like addition) does the opposite of its inverse (like subtraction).
In trigonometry, this idea is the same. Inverse trigonometric functions do the opposite of the normal trigonometric functions. More specifically,
Inverse sine, \(sin^{-1}\) or \(arcsin\), does the opposite of the sine function.
Inverse cosine, \(cos^{-1}\) or \(arccos\) , does the opposite of the cosine function.
Inverse tangent, \(tan^{-1}\) or \(arctan\), does the opposite of the tangent function.
Inverse cotangent, \(cot^{-1}\) or \(arccot\), does the opposite of the cotangent function.
Inverse secant, \(sec^{-1}\) or \(arcsec\), does the opposite of the secant function.
Inverse cosecant, \(csc^{-1}\) or \(arccsc\), does the opposite of the cosecant function.
The inverse trigonometric functions are also called arc functions because, when given a value, they return the length of the arc needed to obtain that value. This is why we sometimes see inverse trig functions written as \(arcsin, arccos, arctan\), etc.
Using the right triangle below, let's define the inverse trig functions!
The inverse trigonometric functions are inverse operations to the trigonometric functions. In other words, they do the opposite of what the trig functions do. In general, if we know a trig ratio but not the angle, we can use an inverse trig function to find the angle. This leads us to define them in the following way:
Trig functions – given an angle, return a ratio | Inverse trig functions – given a ratio, return an angle |
\[\sin(\theta)=\dfrac{opposite}{hypotenuse}\] | \[(\theta)=sin^{-1} \dfrac{opposite}{hypotenuse}\] |
\[\cos(\theta)=\dfrac{adjacent}{hypotenuse}\] | \[(\theta)=cos^{-1}\dfrac{adjacent}{hypotenuse}\] |
\[\tan(\theta)=\dfrac{opposite}{adjacent}\] | \[(\theta)=\tan^{-1}\dfrac{opposite}{adjacent}\] |
\[\cot(\theta)=\dfrac{adjacent}{opposite}\] | \[(\theta)=\cot^{-1}\dfrac{adjacent}{opposite}\] |
\[\sec(\theta)=\dfrac{hypotenuse}{adjacent}\] | \[(\theta)=\sec^{-1}\dfrac{hypotenuse}{adjacent}\] |
\[\csc(\theta)=\dfrac{hypotenuse}{opposite}\] | \[(\theta)=csc^{-1}\dfrac{hypotenuse}{opposite}\] |
As you might have noticed, the notation used to define the inverse trig functions makes it look like they have exponents. While it may seem like it, the \(-1\) superscript is NOT an exponent! In other words, \(\sin^{-1}(x)\) is not the same as \(\dfrac{1}{\sin(x)}\)! The \(-1\) superscript simply means “inverse.”
For perspective, if we were to raise a number or variable to the \(-1\) power, this means we are asking for its multiplicative inverse, or its reciprocal.
So, why are the inverse trig functions any different?
Therefore:
This pattern continues for any function!
The main inverse trigonometric formulas are listed in the table below.
The 6 main inverse trigonometric formulas | |
Inverse sine, or, arc sine: \(y=sin^{-1}(x)=arcsin(x)\) | Inverse cosecant, or, arc cosecant: \(y=csc^{-1}(x)=arccsc(x)\) |
Inverse cosine, or, arc cosine: \(y=cos^{-1}(x)=arccos(x)\) | Inverse secant, or, arc secant: \(y=sec^{-1}(x)=arcsec(x)\) |
Inverse tangent, or, arc tangent: \(y=tan^{-1}(x)=arctan(x)\) | Inverse cotangent, or, arc cotangent: \(y=cot^{-1}(x)=arcot(x)\) |
Let's explore these with an example!
Consider the inverse trigonometric function: \(y=sin^{-1}(x)\)
Based on the definition of inverse trigonometric functions, this implies that: \(sin(y)=x\).
Keeping this in mind, say we want to find the angle θ in the right triangle below. How can we go about doing so?
Solution:
What do the inverse trigonometric functions look like? Let's check out their graphs.
But, before we can graph the inverse trigonometric functions, we need to talk about their domains. Because the trigonometric functions are periodic, and therefore not one-to-one, they don't have inverse functions. So then, how can we have inverse trigonometric functions?
To find inverses of the trigonometric functions, we must either restrict or specify their domains so that they are one-to-one! Doing so allows us to define a unique inverse of either sine, cosine, tangent, cosecant, secant, or cotangent.
In general, we use the following convention when evaluating inverse trigonometric functions:
Inverse trig function | Formula | Domain |
Inverse sine / arc sine | \(y=sin^{-1}(x)=arcsin(x)\) | \([-1,1]\) |
Inverse cosine / arc cosine | \(y=cos^{-1}(x)=arccos(x)\) | \([-1,1]\) |
Inverse tangent / arc tangent | \(y=tan^{-1}(x)=arctan(x)\) | \(-\infty, \infty\) |
Inverse cotangent / arc cotangent | \(y=cot^{-1}(x)=arccot(x)\) | \(-\infty, infty\) |
Inverse secant / arc secant | \(y=sec^{-1}(x)=arcsec(x)\) | \((-\infty, -1] \cup [1, \infty)\) |
Inverse cosecant / arc cosecant | \(y=csc^{-1}(x)=arccsc(x)\) | \((-\infty, -1] \cup [1, \infty)\) |
These are just the conventional, or standard, domain we choose when restricting the domains. Remember, since trig functions are periodic, there are an infinite number of intervals on which they are one-to-one!
To graph the inverse trigonometric functions, we use the graphs of the trigonometric functions restricted to the domains specified in the table above and reflect those graphs about the line \(y=x\), just as we did for finding Inverse Functions.
Below are the 6 main inverse trigonometric functions and their graphs, domain, range (also known as the principal interval), and any asymptotes.
The graph of \(y=sin^{-1}(x)=arcsin(x)\) | The graph of \(y=cos^{-1}(x)=arccos(x)\) | ||
Domain: \([-1,1]\) | Range: \([-\dfrac{\pi}{2},\dfrac{\pi}{2}]\) | Domain: \([-1,1]\) | Range: \([0,\pi]\) |
The graph of \(y=sec^{-1}(x)=arcsec(x)\) | The graph of \(y=csc^{-1}(x)=arccsc(x)\) | ||
Domain: \((-\infty, -1] \cup [1, \infty)\) | Range: \((0, \dfrac{\pi}{2}] \cup [\dfrac{\pi}{2}, \pi)\) | Domain: \((-\infty, -1] \cup [1, \infty)\) | Range: \((- \dfrac{\pi}{2},0] \cup [0,\dfrac{\pi}{2})\) |
Asymptote: \(y=\dfrac{\pi}{2}\) | Asymptote: \(y=0\) |
The graph of \(y=tan^{-1}(x)=arctan(x)\) | The graph of \(y=cot^{-1}(x)=arccot(x)\) | ||
Domain: \(-\infty, \infty\) | Range: \([-\dfrac{\pi}{2},\dfrac{\pi}{2}]\) | Domain: \(-\infty, \infty\) | Range: \(0, \pi\) |
Asymptotes: \(y=-\dfrac{\pi}{2}, y=\dfrac{\pi}{2}\) | Asymptotes: \(y=0, y=\pi\) |
When we deal with inverse trigonometric functions, the unit circle is still a very helpful tool. While we typically think about using the unit circle to solve trigonometric functions, the same unit circle can be used to solve, or evaluate, the inverse trigonometric functions.
Before we get to the unit circle itself, let's take a look at another, simpler tool. The diagrams below can be used to help us remember from which quadrants the inverse trigonometric functions on the unit circle will come.
Just as the cosine, secant, and cotangent functions return values in Quadrants I and II (between 0 and 2π), their inverses, arc cosine, arc secant, and arc cotangent, do as well.
Just as the sine, cosecant, and tangent functions return values in Quadrants I and IV (between \(-\dfrac{\pi}{2}\) and \(\dfrac{\pi}{2}\)), their inverses, arc sine, arc cosecant, and arc tangent, do as well. Note that the values from Quadrant IV will be negative.
These diagrams assume the conventional restricted domains of the inverse functions.
There is a distinction between finding inverse trigonometric functions and solving for trigonometric functions.
Say we want to find \(\sin^{-1}\left( \dfrac{\sqrt{2}}{2} \right)\).
Now, say we want to solve \(\sin(x)=\dfrac{\sqrt{2}}{2}\).
We might recall that we can use the Unit Circle to solve trigonometric functions of special angles: angles that have trigonometric values that we evaluate exactly.
When using the unit circle to evaluate inverse trigonometric functions, there are several things we need to keep in mind:
In calculus, we will be asked to find derivatives and integrals of inverse trigonometric functions. In this article, we present a brief overview of these topics.
For a more in-depth analysis, please refer to our articles on Derivatives of Inverse Trigonometric Functions and Integrals Resulting in Inverse Trigonometric Functions.
A surprising fact about the Derivatives of Inverse Trigonometric Functions is that they are algebraic functions, not trigonometric functions. The derivatives of inverse trigonometric functions are defined as:
\[\dfrac{d}{dx}\sin^{-1}(x)=\dfrac{1}{\sqrt{1-(x)^2}}\]
\[\dfrac{d}{dx}\cos^{-1}(x)=\dfrac{-1}{\sqrt{1+(x)^2}}\]
\[\dfrac{d}{dx}\tan^{-1}(x)=\dfrac{1}{1+(x)^2}\]
\[\dfrac{d}{dx}\cot^{-1}(x)=\dfrac{-1}{1+(x)^2}\]
\[\dfrac{d}{dx}\sec^{-1}(x)=\dfrac{1}{|x|\sqrt{(x)^2-1}}\]
\[\dfrac{d}{dx}\csc^{-1}(x)=\dfrac{-1}{|x|\sqrt{(x)^2-1}}\]
Previously, we have developed the formulas for the derivatives of inverse trigonometric functions. These formulas are what we use to develop the Integrals Resulting in Inverse Trigonometric Functions. These integrals are defined as:
\[\int \dfrac{du}{\sqrt{a^2-u^2}}=\sin^{-1}\left( \dfrac{u}{a} \right)+C\]
\[\int \dfrac{du}{\sqrt{a^2+u^2}}=\dfrac{1}{a}\tan^{-1}\left( \dfrac{u}{a} \right)+C\]
\[\int \dfrac{du}{u \sqrt{a^2+u^2}}=\dfrac{1}{a}\sec^{-1}\left( \dfrac{u}{a} \right)+C\]
There are 6 inverse trigonometric functions, so why are there only three integrals? The reason for this is that the remaining three integrals are just negative versions of these three. In other words, the only difference between them is whether the integrand is positive or negative.
Other than the integrals that result in the inverse trigonometric functions, there are integrals that involve the inverse trigonometric functions. These integrals are:
The inverse trigonometric integrals that involve arc sine.
\(\int sin^{-1} u du = sin^{-1}(u)+\sqrt{1-u^2}+C\)
\(\int u \sin^{-1}u du=\dfrac{2u^2-1}{4} \sin^{-1}(u)+\dfrac{u\sqrt{1-u^2}}{4}+C\)
\(\\int u^n sin^{-1}u du \dfrac{1}{n+1} \left[ u^{n+1} \sin^{-1}(u) - \int \dfrac{u^{n+1}du}{\sqrt{1-u^2}}, n \neq -1 \right]\)
The inverse trigonometric integrals that involve arc cosine.
\(\int cos^{-1}udu =cos^{-1}(u)-\sqrt{1-u^2}+C\)
\(\int cos^{-1} u du = \dfrac{1}{n+1}\left[ u^{n+1} \cos^{-1} (u)+ \int \dfrac{u^{n+1}du}{\sqrt{1-u^2}} \right], n \neq -1\)
The inverse trigonometric integrals that involve arc tangent.
\(\int tan^{-1}udu=tan^{-1}(u)-\dfrac{1}{2}ln(1+u^2)+C\)
\(\int u \tan^{-1} u du = \dfrac{u^2-1}{2}\tan^{-1}(u)+C\)
\(\int u^n tan^{-1} udu = \dfrac{1}{n+1}\left[ \dfrac{u^{n+1} du}{1+u^2}\right], n \neq -1\)
When we solve, or evaluate, inverse trigonometric functions, the answer we get is an angle.
Evaluate \(\cos^{-1} \left( \dfrac{1}{2}\right) \).
Solution:
To evaluate this inverse trig function, we need to find an angle \(\theta\) such that \(\cos(\theta)=\dfrac{1}{2}\).
What about the composition of a trigonometric function and its inverse?
Let's consider the two expressions:
\[\sin\left( sin^{-1}\left( \dfrac{\sqrt{2}}{2} \right) \right)\]
and
\[\sin^{-1}(\sin(\pi))\]
Solutions:
Let's think about the answer for the second expression in the example above.
Isn't the inverse of a function supposed to undo the original function? Why isn't \( \sin^{-1} ( \sin (\pi) )= \pi \)?
Remembering the definition of inverse functions: a function \(f\) and its inverse \(f^{-1}\) satisfy the conditions \( f (f^{-1}(y))=y\)for all y in the domain of \( f^{-1}\) , and \(f^{-1}(f(x))=x\) for all \(x\) in the domain of \(f\).
So, what happened in this example?
Then, what about \(\sin(\sin^{-1}(y))\)? Does this expression have a similar issue?
This expression does not have the same issue because the domain of \(\sin^{-1}\) is the interval \([-1, 1]\).
So, \(\sin(\sin^{-1}(y))=y\) if \(-1 \leq y \leq 1\). This expression is not defined for any other values of \(y\).
Let's summarize these findings:
The conditions for trigonometric functions and their inverses to cancel each other | |
\(\sin(\sin^{-1}(y)=y)\) if \(-1 \leq y \leq 1\) | \(\sin^{-1}(\sin(x))=x\) if \( -\dfrac{\pi}{2}\leq x \leq \dfrac{\pi}{2} \) |
\(\cos(\cos^{-1}(y)=y)\) if \(-1 \leq y \leq 1\) | \(\cos^{-1}(\cos(x))=x\) if \( 0 \leq x \leq \pi \) |
\(\tan(\tan^{-1}(y)=y)\) if \(-\infty \leq y \leq \infty\) | \(\tan^{-1}(\tan(x))=x\) if \( -\dfrac{\pi}{2}\leq x \leq \dfrac{\pi}{2} \) |
\(\cot(\cot^{-1}(y)=y)\) if \(-\infty \leq y \leq \infty\) | \(\cot^{-1}(\cot(x))=x\) if \( 0 < x < \pi \) |
\(\sec(\sec^{-1}(y)=y)\) if \(( -\infty, -1] \leq \cup [1, \infty)\) | \(\sec^{-1}(\sec(x))=x\) if \( 0 < x < \dfrac{\pi}{2} \cup \dfrac{\pi}{2} < x < \pi\) |
\(\csc(\csc^{-1}(y)=y)\) if \(( -\infty, -1] \leq \cup [1, \infty)\) | \(\csc^{-1}(\csc(x))=x\) if \( -\dfrac{\pi}{2} < x < \-0 \cup 0 < x < \dfrac{\pi}{2} \) |
Evaluate the following expressions:
Solutions:
On most graphing calculators, you can directly evaluate inverse trigonometric functions for inverse sine, inverse cosine, and inverse tangent.
When it is not explicitly specified, we restrict the inverse trigonometric functions to the standard bounds specified in the section “inverse trigonometric functions in a table”. We saw this restriction in place in the first example.
However, there may be cases where we want to find an angle corresponding to a trigonometric value evaluated within a different specified bound. In such cases, it is useful to remember the trigonometric quadrants:
Given the following, find \(theta\).
\[\sin(\theta)=-0.625\]
where
\[90^o< \theta < 270^o\]
Solution:
How can we find the inverse of a function, both algebraically and graphically?
What is the key to inverse operations?
The key here is: an operation (like addition) does the opposite of its inverse (like subtraction).
Inverse trigonometric functions do the __ the normal trigonometric functions.
opposite of
While it may seem like it, the -1 superscript is __
not an exponent
How can we have inverse trigonometric functions?
Because the trigonometric functions are periodic, and therefore not one-to-one, they don't have inverse functions.
BUT, if we either restrict or specify their domains so that they are one-to-one we can define a unique inverse of either sine, cosine, tangent, cosecant, secant, or cotangent.
When using the unit circle to evaluate inverse trigonometric functions, there are several things we need to keep in mind:
If the answer is in Quadrant IV, it must be a negative answer (in other words, we go clockwise from the point (1, 0) instead of counterclockwise).
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