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19.1: The functions of arcsin, arccos, and arctan

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    49073
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    The inverse trigonometric functions are the inverse functions of the \(y=\sin x\), \(y=\cos x\), and \(y=\tan x\) functions restricted to appropriate domains. In this section we give a precise definition of these functions.

    The Inverse Tangent Function

    We start with the inverse to the tangent function \(y=\tan(x)\). Recall that the graph of \(y=\tan(x)\) is the following:

    clipboard_ee0857b050f93178adbeaea512ee557ca.png

    It has vertical asymptotes at \(x=\pm\dfrac{\pi}{2}, \pm\dfrac{3\pi}{2}, \pm\dfrac{5\pi}{2}, \dots\). Note, that \(y=\tan(x)\) is not a one-to-one function in the sense of defintion [DEF:1-to-1] on page . (For example, the horizontal line \(y=1\) intersects the graph at \(x=\dfrac{\pi}{4}\), \(x=\dfrac{\pi}{4}\pm\pi\), \(x=\dfrac{\pi}{4}\pm2\pi\), etc.) However, when we restrict the function to the domain \(D=(\dfrac{-\pi}{2},\dfrac{\pi}{2})\) the restricted function is one-to-one, and, for this restricted function, we may take its inverse function.

    Definition: Inverse Tangent or Arctangent

    The inverse of the function \(y=\tan(x)\) with restricted domain \(D=(\dfrac{-\pi}{2},\dfrac{\pi}{2})\) and range \(R=\mathbb{R}\) is called the inverse tangent or arctangent function. It is denoted by

    \[y=\tan^{-1}(x) \quad \text{ or } \quad y= \arctan(x) \quad \iff \quad \tan(y)=x, \quad y\in\left(-\dfrac{\pi}2,\dfrac{\pi}{2}\right) \nonumber \]

    The arctangent reverses the input and output of the tangent function, so that the arctangent has domain \(D=\mathbb{R}\) and range \(R=\left(\dfrac{-\pi}{2},\dfrac{\pi}{2}\right)\). The graph is displayed below.

    clipboard_ec75d84a1784e1eb22143d29ae3168b9e.png

    Warning

    The notation of \(\tan^{-1}(x)\) and \(\tan^2(x)\) is slightly inconsistent, since the exponentiation symbol is used above in two different ways. In fact, \(\tan^{-1}(x)=\arctan(x)\) refers to the inverse function of the \(\tan(x)\) function. However, when we write \(\tan^2(x)\), we mean

    \[\tan^2(x)=(\tan(x))^2=\tan(x)\cdot \tan(x) \nonumber \]

    Therefore, \(\tan^{-1}(x)\) is the inverse function of \(\tan(x)\) with respect to the composition operation, whereas \(\tan^2(x)\) is the square with respect to the usual product in \(\mathbb{R}\). Note also that the inverse function of the tangent with respect to the product in \(\mathbb{R}\) is \(y=\dfrac{1}{\tan(x)}=\cot(x)\), which is the cotangent.

    Observation: Inverse Tangent Function

    The inverse tangent function is an odd function:

    \[\label{EQU:tan-1(-x)} \boxed{\tan^{-1}(-x)=-\tan^{-1}(x)} \nonumber \]

    This can be seen by observing that the tangent \(y=\tan(x)\) is an odd function (that is \(\tan(-x)=-\tan(x)\)), or directly from the symmetry of the graph with respect to the origin \((0,0)\).

    The next example calculates function values of the inverse tangent function.

    Example \(\PageIndex{1}\)

    Recall the exact values of the tangent function from section 17.1:

    \[\begin{array}{c||c|c|c|c|c}
    x & 0=0^{\circ} & \dfrac{\pi}{6}=30^{\circ} & \dfrac{\pi}{4}=45^{\circ} & \dfrac{\pi}{3}=60^{\circ} & \dfrac{\pi}{2}=90^{\circ} \\
    \hline \hline \tan (x) & 0 & \dfrac{\sqrt{3}}{3} & 1 & \sqrt{3} & \text { undef. }
    \end{array} \nonumber \]

    Solution

    From this, we can deduce function values by reversing inputs and outputs, such as:

    \[\begin{aligned} \tan\Big(\dfrac{\pi}{6}\Big)=\dfrac{\sqrt{3}}{3} &\implies & \tan^{-1}\Big(\dfrac{\sqrt{3}}{3} \Big)=\dfrac{\pi}{6} \\ \tan\Big(\dfrac{\pi}{4}\Big)=1 &\implies & \tan^{-1}\big(1 \big)=\dfrac{\pi}{4} \end{aligned} \nonumber \]

    Also, since \(\tan^{-1}(-x)=-\tan^{-1}(x)\), we obtain the inverse tangent of negative numbers.

    \[\begin{aligned} \tan^{-1}(-\sqrt{3})&=&-\tan^{-1}(\sqrt{3})=-\dfrac{\pi}{3} \\ \tan^{-1}(-1)&=&-\tan^{-1}(1)=-\dfrac{\pi}{4} \end{aligned} \nonumber \]

    We may calculate the inverse tangent of specific values with the calculator using the \(\boxed {\text{2nd}}\) and \(\boxed {\text{tan}}\) keys. For example, \(\tan^{-1}(4.3)\approx 1.34\).

    clipboard_e334a20391b35dfe3c5f87b76695fcbb3.png

    Note, that the answer differs, when changing the mode from radians to degree, since \(\tan^{-1}(4.3) \approx 76.9^\circ \approx 1.34\).

    clipboard_e4696e833ba315a1377f10d3f14bc64cb.png

    The function \(y=\sin^{-1}(x)\)

    Next, we define the inverse sine function. For this, we again first recall the graph of the \(y=\sin(x)\) function, and note that it is also not one-to-one.

    clipboard_e4065a164a663940e3015f3e200b14899.png

    However, when restricting the sine to the domain \(\left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right]\), the restricted function is one-to-one. Note furthermore, that when restricting the domain to \(\left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right]\), the range is \([-1,1]\), and therefore we cannot extend this to a larger domain in a way such that the function remains a one-to-one function. We use the domain \(\left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right]\) to define the inverse sine function.

    Definition: Inverse Sine or Arcsine

    The inverse of the function \(y=\sin(x)\) with restricted domain \(D=\left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right]\) and range \(R=[-1,1]\) is called the inverse sine or arcsine function. It is denoted by

    \[y=\sin^{-1}(x) \quad \text{ or } \quad y= \arcsin(x) \quad \iff \quad \sin(y)=x, \quad y\in\left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right] \nonumber \]

    The arcsine reverses the input and output of the sine function, so that the arcsine has domain \(D=[-1,1]\) and range \(R=\left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right]\). The graph of the arcsine is drawn below.

    clipboard_eb3d626f84b959944ce54f4b1af540e9b.png

    Observation: Inverse Tangent Function

    The inverse sine function is odd:

    \[\label{EQU:sin-1(-x)} \boxed{\sin^{-1}(-x)=-\sin^{-1}(x)} \nonumber \]

    This can again be seen by observing that the sine \(y=\sin(x)\) is an odd function (that is \(\sin(-x)=-\sin(x)\)), or alternatively directly from the symmetry of the graph with respect to the origin \((0,0)\).

    We now calculate specific function values of the inverse sine.

    Example \(\PageIndex{2}\)

    We first recall the known values of the sine.

    \[\begin{array}{c||c|c|c|c|c}
    x & 0=0^{\circ} & \dfrac{\pi}{6}=30^{\circ} & \dfrac{\pi}{4}=45^{\circ} & \dfrac{\pi}{3}=60^{\circ} & \dfrac{\pi}{2}=90^{\circ} \\
    \hline \hline \sin (x) & 0 & \dfrac{1}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{\sqrt{3}}{2} & 1
    \end{array} \nonumber \]

    Solution

    These values together with the fact that the inverse sine is odd, that is \(\sin^{-1}(-x)=-\sin^{-1}(x)\), provides us with examples of its function values.

    \[\begin{aligned} \sin^{-1}\Big(\dfrac{\sqrt{2}}{2}\Big) &= \dfrac{\pi}{4}\\\sin^{-1}(1) &=\dfrac{\pi}{2}\\ \sin^{-1}(0) &= 0\\\sin^{-1}\Big(\dfrac{-1}{2}\Big) &=-\sin^{-1}\Big(\dfrac{1}{2}\Big)=-\dfrac{\pi}{6}\end{aligned} \nonumber \]

    Note, that the domain of \(y=\sin^{-1}(x)\) is \(D=[-1,1]\), so that input numbers that are not in this interval give undefined outputs of the inverse sine:

    \[\sin^{-1}(3) \text{ is undefined} \nonumber \]

    Input values that are not in the above table may be found with the calculator via the \(\boxed {\text{2nd}}\)\(\boxed {\text{sin}}\) keys. We point out, that the output values depend on wether the calculator is set to radian or degree mode. (Recall that the \(\boxed {\text{mode}}\) may be changed via the key).

    clipboard_ed8a7b2e5616f478d1497d273cd9e70be.png

    The function \(y=\cos^{-1}(x)\)

    Finally, we define the inverse cosine. Recall the graph of \(y=\cos(x)\), and note again that the function is not one-to-one.

    clipboard_ed3d50e127849065be95a34bfbf068b0d.png

    In this case, the way to restrict the cosine to a one-to-one function is not as clear as in the previous cases for the sine and tangent. By convention, the cosine is restricted to the domain \([0,\pi]\). This provides a function that is one-to-one, which is used to define the inverse cosine.

    Definition: Inverse Cosine or Arccosine

    The inverse of the function \(y=\cos(x)\) with restricted domain \(D=[0,\pi]\) and range \(R=[-1,1]\) is called the inverse cosine or arccosine function. It is denoted by

    \[y=\cos^{-1}(x) \quad \text{ or } \quad y= \arccos(x) \quad \iff \quad \cos(y)=x,\quad y\in [0,\pi] \nonumber \]

    The arccosine reverses the input and output of the cosine function, so that the arccosine has domain \(D=[-1,1]\) and range \(R=[0,\pi]\). The graph of the arccosine is drawn below.

    clipboard_e74708f181d80c431b1f5f2bc632b8995.png

    Observation: Inverse Cosine

    The inverse cosine function is neither even nor odd. That is, the function \(\cos^{-1}(-x)\) cannot be computed by simply taking \(\pm\cos^{-1}(x)\). But it does have some symmetry given algebraically by the more complicated relation

    \[\label{EQU:cos-1(-x)} \boxed{\cos^{-1}(-x)=\pi-\cos^{-1}(x)} \nonumber \]

    Proof

    We can see that if we shift the graph down by \(\dfrac{\pi}{2}\) the resulting function is odd. That is to say the function with the rule \(\cos^{-1}(x)-\dfrac{\pi}{2}\) is odd:

    \[\cos^{-1}(-x)-\dfrac{\pi}{2}=-(\cos^{-1}(x)-\dfrac{\pi}{2}) \nonumber\]

    which yields \(\ref{EQU:cos-1(-x)}\) upon distributing and adding \(\dfrac{\pi}{2}\).

    Another, more formal approach is as follows. The bottom right relation of [EQU:basic-trig-eqns-wrt-pi] on page states, that we have the relation \(\cos(\pi-y)=-\cos(y)\) for all \(y\). Let \(-1\leq x\leq 1\), and denote by \(y=\cos^{-1}(x)\). That is \(y\) is the number \(0\leq y\leq \pi\) with \(\cos(y)=x\). Then we have

    \(-x=-\cos(y)=\cos(\pi-y) \) (by equation 17.1.2)

    Applying \(\cos^{-1}\) to both sides gives:

    \[\cos^{-1}(-x)=\cos^{-1}(\cos(\pi-y))=\pi-y \nonumber \]

    The last equality follows, since \(\cos\) and \(\cos^{-1}\) are inverse to each other, and \(0\leq y\leq \pi\), so that \(0\leq \pi-y\leq \pi\) are also in the range of the \(\cos^{-1}\). Rewriting \(y=\cos^{-1}(x)\) gives the wanted result:

    \[\cos^{-1}(-x)=\pi-\cos^{-1}(x) \nonumber \]

    This is the equation \(\ref{EQU:cos-1(-x)}\) which we wanted to prove.

    We now calculate specific function values of the inverse cosine.

    Example \(\PageIndex{3}\)

    We first recall the known values of the cosine.

    \[\begin{array}{c||c|c|c|c|c}
    x & 0=0^{\circ} & \dfrac{\pi}{6}=30^{\circ} & \dfrac{\pi}{4}=45^{\circ} & \dfrac{\pi}{3}=60^{\circ} & \dfrac{\pi}{2}=90^{\circ} \\
    \hline \hline \cos (x) & 1 & \dfrac{\sqrt{3}}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{1}{2} & 0
    \end{array} \nonumber \]

    Solution

    Here are some examples for function values of the inverse cosine.

    \[\cos^{-1}\Big(\dfrac{\sqrt{3}}{2}\Big)=\dfrac{\pi}{6}, \quad\quad \cos^{-1}(1)=0,\quad\quad \cos^{-1}(0) =\dfrac{\pi}{2} \nonumber \]

    Negative inputs to the arccosine can be calculated with equation \(\ref{EQU:cos-1(-x)}\), that is \(\cos^{-1}(-x)=\pi-\cos^{-1}(x)\), or by going back to the unit circle definition.

    \[\begin{aligned} \cos^{-1}\Big(-\dfrac{1}{2}\Big)&=\pi-\cos^{-1}\Big(\dfrac{1}{2}\Big)=\pi-\dfrac{\pi}{3}=\dfrac{3\pi-\pi}{3}=\dfrac{2\pi}{3}\\ \cos^{-1}(-1)&=\pi-\cos^{-1}(1)=\pi-0=\pi\end{aligned} \nonumber \]

    Furthermore, the domain of \(y=\cos^{-1}(x)\) is \(D=[-1,1]\), so that input numbers not in this interval give undefined outputs of the inverse cosine.

    \[\cos^{-1}(17) \text{ is undefined} \nonumber \]

    Other input values can be obtained with the calculator by pressing the \(\boxed {\text{2nd}}\)\(\boxed {\text{cos}}\) keys. For example, we obtain the following function values (here using radian measure).

    clipboard_e2499a6ff243240cf59348896af4a8f03.png


    This page titled 19.1: The functions of arcsin, arccos, and arctan is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.