The unit circle, the radian measure and the concept of modulo
The unit circle
Unit circle
The unit circle is the collection of points on the xy -plane which have a distance of 1 from the origin.
A point \left(a,b\right) is on the unit circle if and only if:
a^2+b^2=1
Observe the following:
\left(\dfrac{\sqrt{2}}{2}\right)^2+\left(\dfrac{\sqrt{2}}{2}\right)^2=\dfrac{2}{4}+\dfrac{2}{4}=1
Therefore the following point is on the unit circle:
\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)
Suppose a,b,c denote the lengths of the sides of a right triangle as illustrated in the following image:
The Pythagorean Theorem implies the following:
a^2+b^2=c^2
The unit circle has 1 for diameter.
It is for this reason that if \left(a,b\right) is on the unit circle then a^2+b^2=1.
If \left(a,b\right) is on the unit circle then so are the following points:
- \left(-a,b\right)
- \left(a,-b\right)
- \left(-a,-b\right)
The following point is on the unit circle:
\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)
It follows that the following points are also on the unit circle:
- \left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)
- \left(\dfrac{\sqrt{2}}{2}, -\dfrac{\sqrt{2}}{2}\right)
- \left(-\dfrac{\sqrt{2}}{2}, -\dfrac{\sqrt{2}}{2}\right)
If a point \left(a,b\right) is on the unit circle, then so is the point \left(b,a\right).
Observe the following:
\left(\dfrac{\sqrt{3}}{2}\right)^2+\left(\dfrac{1}{2}\right)^2=\dfrac{3}{4}+\dfrac{1}{4}=1
Therefore the point \left(\dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right) is on the unit circle. It follows that \left(\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right) is also on the unit circle.
Hence the following points are also on the unit circle:
- \left(-\dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right)
- \left(\dfrac{\sqrt{3}}{2},-\dfrac{1}{2}\right)
- \left(\dfrac{-\sqrt{3}}{2},-\dfrac{1}{2}\right)
- \left(-\dfrac{1}{2},-\dfrac{\sqrt{3}}{2}\right)
- \left(-\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right)
- \left(\dfrac{1}{2},-\dfrac{\sqrt{3}}{2}\right)
The radian measure
The unit circle provides a natural way to measure the degree of an angle. To measure an angle formed by two rays, place a unit circle at the vertex of the angle with one of the rays passing through the point \left(1{,}0\right) on the circle.
The two rays will cut a portion of the circumference of the unit circle. The length of that portion measured counterclockwise is the radian measure of the angle.
In the following image, an angle of measure \theta radians is given. The length of the arc colored in blue is exactly \theta units.
If a circle has radius r then the circumference of that circle has radius 2\pi r. In particular, the unit circle has a circumference of length 2\pi. Therefore the measure of every angle in radians is between 0 and 2\pi.
We interpret positive radians as traveling counterclockwise on the unit circle and and negative radians as traveling clockwise.
The following image indicates the angle of measure in radians of distinguished points on the unit circle.
The concept of modulo
If a particle is on the unit circle and travels a distance of 2\pi exactly along the unit circle in a counterclockwise direction then the particle will stop in the exact same position it started.
When measuring angles in radians, we are not limited to measurements between 0 and 2\pi as we can allow for any positive number. If \theta is a measurement of an angle in radians then any other radian measurement of the form \theta+2n\pi where n is an integer produces the same angle as \theta.
The point \left(\dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right) on the unit circle corresponds to a radian measure of \dfrac{\pi}{6}. But the point \left(\dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right) also corresponds to the following radian measures:
- \dfrac{\pi}{6}+2\pi=\dfrac{13\pi}{6}
- \dfrac{\pi}{6}+4\pi=\dfrac{25\pi}{6}
- \dfrac{\pi}{6}-2\pi=\dfrac{-11\pi}{6}
If \theta_1 and \theta_2 are radian measurements then \theta_1 and \theta_2 correspond to the same point on the unit circle if and only if \theta_1-\theta_2 is an integer multiple of 2\pi.
If this is the case, then we say that \theta_1 and \theta_2 are the same modulo 2\pi.
Observe the following:
\dfrac{19\pi}{4}-\dfrac{3\pi}{4}=\dfrac{16\pi}{4}=4\pi=2\cdot 2\pi
Therefore \dfrac{19\pi}{4} and \dfrac{3\pi}{4} are the same modulo 2\pi and the radian measurements of \dfrac{19\pi}{4} and \dfrac{3\pi}{4} both correspond to the point \left(-\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right) on the unit circle.
\cos\left(x\right), \sin \left(x\right), and \tan\left(x\right)
This lesson will discuss the properties of many new functions. The most important of those functions that we will introduce are the functions \cos\left(x\right), \sin\left(x\right), and \tan\left(x\right). All other functions in this lesson will be variants of these three functions and their properties can be deduced from these three functions.
\cos\left(x\right)
Definition
The cosine function
For any real number x let \cos\left(x\right) be the x -value on the unit circle which corresponds to a measurement of x radians.
- Positive x -values correspond to counter clockwise movement on the unit circle and negative x -values correspond to clockwise movement on the unit circle. It follows that the domain of the cosine function is all real numbers.
- The diameter of the unit circle is 1 and the center is the origin, hence the range of the cosine function is \left[-1{,}1\right].
The following table is a list of distinguished values of the cosine function:
| x | \cos\left(x\right) |
| 0 | 1 |
| \dfrac{\pi}{6} | \dfrac{\sqrt{3}}{2} |
| \dfrac{\pi}{4} | \dfrac{\sqrt{2}}{2} |
| \dfrac{\pi}{3} | \dfrac{1}{2} |
| \dfrac{\pi}{2} | 0 |
| \dfrac{2\pi}{3} | \dfrac{-1}{2} |
| \dfrac{3\pi}{4} | \dfrac{-\sqrt{2}}{2} |
| \dfrac{5\pi}{6} | \dfrac{-\sqrt{3}}{2} |
| \pi | -1 |
| \dfrac{7\pi}{6} | \dfrac{-\sqrt{3}}{2} |
| \dfrac{5\pi}{4} | \dfrac{-\sqrt{2}}{2} |
| \dfrac{4\pi}{3} | \dfrac{-1}{2} |
| \dfrac{3\pi}{2} | 0 |
| \dfrac{5\pi}{3} | \dfrac{1}{2} |
| \dfrac{7\pi}{4} | \dfrac{\sqrt{2}}{2} |
| \dfrac{11\pi}{6} | \dfrac{\sqrt{3}}{2} |
Graphic representation, domain and range
The following is the graph of the cosine function:
The domain of \cos\left(x\right) is all real numbers and the range of \cos\left(x\right) is \left[-1{,}1\right].
Axis and points of symmetry
The cosine function is an even function. That is:
- For any real number x, \cos\left(x\right)=\cos\left(-x\right)
- The graph of \cos\left(x\right) is symmetric about the y -axis
By examining the unit circle, we notice that it is natural that the cosine function is an even function. For every real number x we have:
\cos\left(-x\right)=\cos\left(x\right)
Period and amplitude
The function \cos\left(x\right) is periodic with period 2\pi.
Two angles measured in radians correspond to equivalent points on the unit if the two measurements are the same modulo 2\pi. Therefore, for any real number x:
\cos\left(x+2\pi\right)=\cos\left(x\right)
The maximal value of \cos\left(x\right) is 1. Therefore the amplitude of \cos\left(x\right) is 1.
Variation and sign table
The graph of \cos\left(x\right) has x -intercepts at all points of the form \left(\dfrac{\pi}{2}+2n\pi,0\right) and \left(\dfrac{3\pi}{2}+2n\pi,0\right) where n can be any integer. The following sign chart indicates on which intervals \cos\left(x\right) is positive and negative:
The following is the graph of \cos\left(x\right). The portions of the graph where \cos\left(x\right) is positive are in red and the portions of the graph where \cos\left(x\right) is negative are in blue.
The cosine function:
- Increases on intervals of the form \left(\pi+2k\pi,2\pi+2k\pi\right), k being an integer.
- Decreases on intervals of the form \left(2k\pi, 3k\pi\right), k being an integer.
\sin\left(x\right)
Definition
The sine function
For any real number x let \sin\left(x\right) be the y -value on the unit circle which corresponds to a measurement of x radians.
The following table is a list of distinguished values of the sine function:
| x | \sin\left(x\right) |
| 0 | 0 |
| \dfrac{\pi}{6} | \dfrac{1}{2} |
| \dfrac{\pi}{4} | \dfrac{\sqrt{2}}{2} |
| \dfrac{\pi}{3} | \dfrac{\sqrt{3}}{2} |
| \dfrac{\pi}{2} | 1 |
| \dfrac{2\pi}{3} | \dfrac{\sqrt{3}}{2} |
| \dfrac{3\pi}{4} | \dfrac{\sqrt{2}}{2} |
| \dfrac{5\pi}{6} | \dfrac{1}{2} |
| \pi | 0 |
| \dfrac{7\pi}{6} | \dfrac{-1}{2} |
| \dfrac{5\pi}{4} | \dfrac{-\sqrt{2}}{2} |
| \dfrac{4\pi}{3} | \dfrac{-\sqrt{3}}{2} |
| \dfrac{3\pi}{2} | -1 |
| \dfrac{5\pi}{3} | \dfrac{-\sqrt{3}}{2} |
| \dfrac{7\pi}{4} | \dfrac{-\sqrt{2}}{2} |
| \dfrac{11\pi}{6} | \dfrac{-1}{2} |
Graphic representation, domain and range
The following is the graph of the sine function:
The domain of \sin\left(x\right) is all real numbers and the range of \sin\left(x\right) is \left[-1{,}1\right].
Axis and points of symmetry
The sine function is an odd function. That is:
- For any real number x, \sin\left(-x\right)=-\sin\left(x\right)
- The graph of \sin\left(x\right) is symmetric about the origin.
By examining the unit circle, we observe that for any real number x:
\sin\left(-x\right)=-\sin\left(x\right)
Therefore, \sin\left(x\right) is an odd function.
Period and amplitude
Two angles measured in radians correspond to equivalent points on the unit if the two measurements are the same modulo 2\pi. Therefore the function \sin\left(x\right) is periodic with period 2\pi.
The periodicity of the sine function is apparent when observing the unit circle. A particle on the unit circle traveling a distance of x will stop on the same spot of the unit circle as a particle that traveled a distance of x+2\pi.Therefore, for any real number x:
\sin\left(x+2\pi\right)=\sin\left(x\right)
The maximal value of \sin\left(x\right) is 1. Therefore the amplitude of \sin\left(x\right) is 1.
Variation and sign table
The graph of \sin\left(x\right) has x -intercepts at all points of the form \left(n\pi,0\right) where n can be any integer. The following sign chart indicates on which intervals \sin\left(x\right) is positive and negative:
The following is the graph of \sin\left(x\right). The portions of the graph where \sin\left(x\right) is positive are in red and the portions of the graph where \sin\left(x\right) is negative are in blue.
The sine function:
- Increases on intervals of the form \left(\dfrac{-\pi}{2}+2k\pi, \dfrac{\pi}{2}+2k\pi\right), where k is an integer.
- Decreases on intervals of the form \left(\dfrac{\pi}{2}+2k\pi, \dfrac{3\pi}{2}+2k\pi\right), where k is an integer.
\tan\left(x\right)
Definition
The tangent function
For any real number x such that \cos\left(x\right)\not = 0 we have:
\tan\left(x\right)=\dfrac{\sin\left(x\right)}{\cos\left(x\right)}
The following table is a list of distinguished values of the tangent function:
| x | \tan\left(x\right) |
| 0 | 0 |
| \dfrac{\pi}{6} | \dfrac{\sqrt{3}}{3} |
| \dfrac{\pi}{4} | 1 |
| \dfrac{\pi}{3} | \sqrt{3} |
| \dfrac{\pi}{2} | undefined |
| \dfrac{2\pi}{3} | -\sqrt{3} |
| \dfrac{3\pi}{4} | -1 |
| \dfrac{5\pi}{6} | \dfrac{-\sqrt{3}}{3} |
| \pi | 0 |
| \dfrac{7\pi}{6} | \dfrac{\sqrt{3}}{3} |
| \dfrac{5\pi}{4} | 1 |
| \dfrac{4\pi}{3} | \sqrt{3} |
| \dfrac{3\pi}{2} | undefined |
| \dfrac{5\pi}{3} | -\sqrt{3} |
| \dfrac{7\pi}{4} | -1 |
| \dfrac{11\pi}{6} | \dfrac{-\sqrt{3}}{3} |
Graphic representation, domain and range
The following is the graph of the tangent function:
- The domain of \tan\left(x\right) is all real numbers except real numbers of the form \dfrac{\pi}{2}+n\pi where n is any integer.
- The range of \tan\left(x\right) is all real numbers.
Axis and points of symmetry
The tangent function is an odd function. That is:
- For any real number x in the range, \tan\left(-x\right)=-\tan\left(x\right)
- The graph of \tan\left(x\right) is symmetric about the origin.
Period and amplitude
The function \tan\left(x\right) is periodic with period \pi. In the graph of \tan\left(x\right) notice that the graph repeats itself between the vertical asymptotes.
The function \tan\left(x\right) does not have a maximal or minimal value. Therefore the function \tan\left(x\right) has infinite amplitude.
Variation and sign table
The graph of \tan\left(x\right) has x -intercepts at all points of the form \left(n\pi,0\right) where n can be any integer.
The graph of \tan\left(x\right) has an x -intercept at \left(0{,}0\right) because:
\tan\left(0\right)=0
The graph of \tan\left(x\right) has vertical asymptotes at:
x=\dfrac{\pi}{2}+n\pi
Where n can be any integer.
The line x=\dfrac{\pi}{2} is a vertical asymptote of \tan\left(x\right).
The following sign chart indicates on which intervals \tan\left(x\right) is positive and negative:
In the following graph of \tan\left(x\right) the portions of the graph of \tan\left(x\right) corresponding to positive values are in red and the portions of the graph of \tan\left(x\right) corresponding to negative values are in blue.
The function \tan\left(x\right) is increasing at every point in its domain.
\csc\left(x\right),\sec\left(x\right),, and \cot\left(x\right)
The properties of the functions \csc\left(x\right),\sec\left(x\right), and \cot\left(x\right) are derived from properties of the functions \cos\left(x\right), \sin \left(x\right) and \tan\left(x\right).
\csc\left(x\right)
The cosecant function
For any real number x such that \sin\left(x\right)\not = 0 we have:
\csc\left(x\right)=\dfrac{1}{\sin\left(x\right)}
Graphic representation, domain and range
- The domain of \csc\left(x\right) is all real numbers except numbers of the form n\pi where n is any integer.
- The range of \csc\left(x\right) is \left(-\infty, -1\left]\cup \right[1,\infty\right).
The following graph is the graph of \csc\left(x\right).
The graph of \csc\left(x\right) has vertical asymptotes of x=n\pi where n can be any integer.
Axis and points of symmetry
The function \sin\left(x\right) is an odd function. Therefore \csc\left(x\right)=\dfrac{1}{\sin\left(x\right)} is an odd function as well.
The cosecant function is an odd function. That is :
- For any real number x in the range, \csc\left(-x\right)=-\csc\left(x\right)
- The graph of \csc\left(x\right) is symmetric about the origin.
Period and amplitude
The function \sin\left(x\right) is periodic with period 2\pi. Therefore \csc\left(x\right)=\dfrac{1}{\sin\left(x\right)} is also periodic with period 2\pi.
The function \csc\left(x\right) is periodic with period 2\pi.
The function \csc\left(x\right) does not have a maximal or minimal value. Therefore the function \csc\left(x\right) has infinite amplitude.
Variation and sign table
The graph of \csc\left(x\right) has no x -intercepts and has vertical asymptotes at x=n\pi where n can be any integer. The following sign chart indicates on which intervals \csc\left(x\right) is positive and negative:
The following is a the graph of \csc\left(x\right). The pieces of the graph corresponding to positive values are in red and the pieces of the graph corresponding to negative values are in blue.
The cosecant function increases on intervals of the following two forms:
- \left(\dfrac{\pi}{2}+2k\pi, \pi+2k\pi\right)
- \left(\pi+2k\pi, \dfrac{3\pi}{2}+2k\pi\right)
where k can be any integer.
The cosecant function decreases on intervals of the following two forms:
- \left(2k\pi, \dfrac{\pi}{2}+2k\pi\right)
- \left(\dfrac{3\pi}{2}+2k\pi, 2\pi +2k\pi\right)
where k can be any integer.
\sec\left(x\right)
The secant function
For any real number x such that \cos\left(x\right)\not = 0 let:
\sec\left(x\right)=\dfrac{1}{\cos\left(x\right)}
Graphic representation, domain and range
- The domain of \sec\left(x\right) is all real numbers except numbers of the form \dfrac{\pi}{2}+n\pi where n is any integer.
- The range of \sec\left(x\right) is \left(-\infty, -1\left]\cup \right[1,\infty\right).
The following is the graph of \sec\left(x\right).
Axis and points of symmetry
The function \cos\left(x\right) is an even function. Therefore the function \sec\left(x\right)=\dfrac{1}{\cos\left(x\right)} is an even function as well.
The secant function is an even function. That is:
- For any x in the range, \sec\left(-x\right)=\sec\left(x\right)
- The graph of \sec\left(x\right) is symmetric about the y -axis.
Period and amplitude
The function \cos\left(x\right) is periodic with period 2\pi. Therefore \sec\left(x\right)=\dfrac{1}{\cos\left(x\right)} is periodic with period 2\pi.
The function \sec\left(x\right) is periodic with period 2\pi.
The function \sec\left(x\right) does not have a maximal or minimal value. Therefore the function \sec\left(x\right) has infinite amplitude.
Variation and sign table
The graph of \sec\left(x\right) has no x -intercepts and has vertical asymptotes at x=\dfrac{\pi}{2}+n\pi where n can be any integer. The following sign chart indicates on which intervals \sec\left(x\right) is positive and negative:
The following is the graph of \sec\left(x\right). The portions of the graph corresponding to positive values are in red and the portions of the graph corresponding to negative values are in blue.
The secant function increases on intervals of the following form:
- \left(2k\pi, \dfrac{\pi}{2}+2k\pi\right)
- \left(\dfrac{\pi}{2}+2k\pi, \pi+2k\pi\right)
where k can be any integer.
The secant function decreases on intervals of the following form:
- \left(\pi+2k\pi, \dfrac{3\pi}{2}+2k\pi\right)
- \left(\dfrac{3\pi}{2}+2k\pi, 2\pi+2k\pi\right)
where k can be any integer.
\cot\left(x\right)
The cotangent function
For any real number x such that \cos\left(x\right)\not = 0 let:
\cot\left(x\right)=\dfrac{\cos\left(x\right)}{\sin\left(x\right)}
Observe that:
\cot\left(x\right)=\dfrac{1}{\tan\left(x\right)}
Graphic representation, domain and range
- The domain of \cot\left(x\right) is all real numbers except numbers of the form n\pi where n is any integer.
- The range of \cot\left(x\right) is all real numbers.
The following is the graph of \cot\left(x\right).
Axis and points of symmetry
The cotangent function is an odd function. That is:
- For any real number x in the range, \cot\left(-x\right)=-\cot\left(x\right)
- The graph of \cot\left(x\right) is symmetric about the origin.
Period and amplitude
The function \cot\left(x\right) is periodic with period \pi.
The function \cot\left(x\right) does not have a maximal or minimal value. Therefore the function \cot\left(x\right) has infinite amplitude.
Variation and sign table
The graph of \cot\left(x\right) has x -intercepts at points of the form \left(\dfrac{\pi}{2}+n\pi,0\right) where n is any integer. The graph of \cot\left(x\right) has vertical asymptotes at x=n\pi where n can be any integer. The following sign chart indicates on which intervals \cot\left(x\right) is positive and negative:
The following is the graph of \cot\left(x\right) with the portions of the graph corresponding to positive values in red and the portions of the graph corresponding to negative values are in blue.
The function \cot\left(x\right) is decreasing at all points in its domain.
\arccos\left(x\right), \arcsin \left(x\right), and \arctan\left(x\right)
The graphs of trigonometric functions do not pass the horizontal line test and therefore the trigonometric functions do not have inverse functions. However, by restricting the domain of a trigonometric function, the graph will then pass the horizontal line test and the function with the restricted domain will have an inverse function.
\arccos\left(x\right)
Definition
The graph of \cos\left(x\right) passes the horizontal line test when the domain of \cos\left(x\right) is restricted to \left[0,\pi\right].
Arccosine
Let \arccos\left(x\right) be the inverse function of \cos\left(x\right) with the restricted domain of \left[0,\pi\right].
The following table is a list of distinguished values of \arccos\left(x\right) :
| x | \arccos\left(x\right) |
| -1 | \pi |
| \dfrac{-\sqrt{3}}{2} | \dfrac{5\pi}{6} |
| \dfrac{-\sqrt{2}}{2} | \dfrac{3\pi}{4} |
| \dfrac{-1}{2} | \dfrac{2\pi}{3} |
| 0 | \dfrac{\pi}{2} |
| \dfrac{1}{2} | \dfrac{\pi}{3} |
| \dfrac{\sqrt{2}}{2} | \dfrac{\pi}{4} |
| \dfrac{\sqrt{3}}{2} | \dfrac{\pi}{6} |
| 1 | 0 |
Graphic representation, domain and range
The graph of \arccos\left(x\right) is obtained by reflecting the graph of \cos\left(x\right) with a restricted domain of \left[0,\pi\right] across the line y=x.
- The domain of \arccos\left(x\right) is \left[-1{,}1\right].
- The range of \arccos\left(x\right) is \left[0,\pi\right].
Axis and points of symmetry
The arccosine function is symmetric about the point \left(0,\dfrac{\pi}{2}\right). Specifically, this means that if the graph of \arccos\left(x\right) is reflected across the y-axis and the line y=\dfrac{\pi}{2} then the resulting image will again be the graph of \arccos\left(x\right).
Period and amplitude
The function \arccos\left(x\right) is not periodic.
The maximal value of \arccos\left(x\right) is \pi. Therefore the function \arccos\left(x\right) has an amplitude of \pi.
Variation and sign table
The function \arccos\left(x\right) is nonnegative for all values of x in its domain.
The function \arccos\left(x\right) is decreasing at all points in its domain.
\arcsin\left(x\right)
Definition
The graph of \sin\left(x\right) passes the horizontal line test when the domain of \sin\left(x\right) is restricted to \left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right].
Arcsine
Let \arcsin\left(x\right) be the inverse function of \sin\left(x\right) with the restricted domain of \left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right].
The following table is a list of distinguished values of \arcsin\left(x\right) :
| x | \arcsin\left(x\right) |
| -1 | \dfrac{-\pi}{2} |
| \dfrac{-\sqrt{3}}{2} | \dfrac{-\pi}{3} |
| \dfrac{-\sqrt{2}}{2} | \dfrac{-\pi}{4} |
| \dfrac{-1}{2} | \dfrac{-\pi}{6} |
| 0 | 0 |
| \dfrac{1}{2} | \dfrac{\pi}{6} |
| \dfrac{\sqrt{2}}{2} | \dfrac{\pi}{4} |
| \dfrac{\sqrt{3}}{2} | \dfrac{\pi}{3} |
| 1 | \dfrac{\pi}{2} |
Graphic representation, domain and range
The graph of \arcsin\left(x\right) is obtained by reflecting the graph of \sin\left(x\right) with a restricted domain of \left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right] across the line y=x.
- The domain of \arcsin\left(x\right) is \left[-1{,}1\right].
- The range of \arcsin\left(x\right) is \left[0,\pi\right].
Axis and points of symmetry
The arcsine function is an odd function. That is
- For every x -value in the domain of \arcsin\left(x\right), \arcsin\left(-x\right)=-\arcsin\left(x\right)
- The graph of \arcsin\left(x\right) is symmetric about the origin.
Period and amplitude
The function \arcsin\left(x\right) is not periodic.
The maximal value of \arcsin\left(x\right) is \dfrac{\pi}{2}. Therefore \arcsin\left(x\right) has an amplitude of \dfrac{\pi}{2}.
Variation and sign table
The function \arcsin\left(x\right) takes on negative values whenever -1\leq x\lt 0 and takes on positive values whenever 0\lt x\leq 1.
The following is the graph of \arcsin\left(x\right) with the portion of the graph corresponding to positive values in red and the portion of the graph corresponding to negative values in blue.
The function \arcsin\left(x\right) is increasing at all points in its domain.
\arctan\left(x\right)
Definition
The graph of \tan\left(x\right) passes the horizontal line test when the domain of \tan\left(x\right) is restricted to \left(\dfrac{-\pi}{2},\dfrac{\pi}{2}\right).
Arctangent
Let \arctan\left(x\right) be the inverse function of \tan\left(x\right) with the restricted domain of \left(\dfrac{-\pi}{2},\dfrac{\pi}{2}\right).
The following table is a list of distinguished values of \arctan\left(x\right) :
| x | \arctan\left(x\right) |
| -\sqrt{3} | \dfrac{-\pi}{3} |
| -1 | \dfrac{-\pi}{4} |
| \dfrac{-\sqrt{3}}{3} | \dfrac{-\pi}{6} |
| 0 | 0 |
| \dfrac{\sqrt{3}}{3} | \dfrac{\pi}{6} |
| 1 | \dfrac{\pi}{4} |
| \sqrt{3} | \dfrac{\pi}{3} |
Graphic representation, domain and range
The graph of \arctan\left(x\right) is obtained by reflecting the graph of \tan\left(x\right) with a restricted domain of \left(\dfrac{-\pi}{2},\dfrac{\pi}{2}\right) across the line y=x.
- The domain of \arctan\left(x\right) is all real numbers.
- The range of \arctan\left(x\right) is \left(\dfrac{-\pi}{2},\dfrac{\pi}{2}\right).
Axis and points of symmetry
The arctangent function is an odd function. That is:
- \arctan\left(-x\right)=-\arctan\left(x\right) for every real number x in the domain of \arctan\left(x\right).
- The graph of \arctan\left(x\right) is symmetric about the origin.
Period and amplitude
The function \arctan\left(x\right) is not periodic.
The function \arctan\left(x\right) approaches the line y=\dfrac{\pi}{2} as x tends towards \infty, and approaches the line y=\dfrac{-\pi}{2} as x tends towards -\infty. In particular, the amplitude of \arctan\left(x\right) is \dfrac{\pi}{2}.
Variation and sign table
The function \arctan\left(x\right) is positive whenever x\gt 0 and is negative whenever x\lt 0.
The following is the graph of \arctan\left(x\right) with the portion of the graph corresponding to positive values in red and the portion of the graph corresponding to negative values in blue.
The function \arctan\left(x\right) is increasing at each point in its domain.