Precalculus

Trigonometry

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Graphs of Sine and Cosine Functions

Definitions of Sine and Cosine

The sine and cosine functions are defined in terms of the coordinate of points lying on the unit circle centered at the origin.

Consider a point \(M\left( {x,y} \right)\) on the unit circle. Suppose that the radius \(r = OM\) makes an angle \(t\) with the positive direction of the \(x-\)axis.

Definition of sine and cosine functions.
Figure 1.

The input of the functions \(\sin t\) and \(\cos t\) is the measure of the angle \(t.\) The output of the sine function \(\sin t\) is the vertical coordinate of the point \(M.\) Respectively, the output of the cosine function \(\cos t\) is the horizontal coordinate of the point \(M.\)

Thus, the coordinates of the point \(M\left( {x,y} \right)\) are determined by the functions

\[x = \cos t,\;y = \sin t,\]

depending on the angle \(t.\)

The Graph and Properties of the Sine Function

Graph of the Sine Function \(y = \sin t\)

When \(t = 0,\) the point \(M\) has coordinates \(M\left( {1,0} \right),\) that is,

\[\sin \left( {t = 0} \right) = 0.\]

When the point \(M\) is rotated counterclockwise about the origin \(90\) degrees from \(t = 0\) to \(t = \frac{\pi }{2},\) the value of sine increases from \(0\) to \(1.\) As the angle goes from \(t = \frac{\pi }{2}\) to \(t = \pi ,\) the value of \(\sin t\) decreases from \(1\) to \(0.\) Then as the radius \(r = OM\) enters the third and fourth quadrant, \(\sin t\) becomes negative, and first decreases from \(0\) to \(-1\), and then increases from \(-1\) to \(0.\) All of these changes are shown in Figure \(2\) below.

Graph of sine function
Figure 2.

Domain and Codomain of the Sine Function

The sine function is defined for all real numbers \(t\) and take all values from the closed interval \(\left[ { - 1,1} \right].\) Hence,

\[\text{dom}\left( {\sin t} \right) = \mathbb{R},\;\text{codom}(\sin t) = \left[ { - 1,1} \right],\]

where dom denotes the domain of the function, and codom denotes its codomain or range.

Parity of the Sine Function

The sine function is an odd function:

\[\sin \left( { - t} \right) = - \sin t.\]

It is symmetric about the origin as seen above from the graph.

Periodicity of the Sine Function

The sine function is periodic with the least period \(2\pi:\)

\[\sin \left( {t + 2\pi n} \right) = \sin t,\]

where \(n \in \mathbb{Z}.\)

Zeros of the Sine Function

The solutions of the equation \(\sin t = 0\) are given by

\[t = \pi n,\;n \in \mathbb{Z}.\]

Extreme Values of the Sine Function

The sine function reaches its maximum value \(\sin t = 1\) at

\[{t_{\max }} = \frac{\pi }{2} + 2\pi n,\;n \in \mathbb{Z}.\]

The minimum value \(\sin t = -1\) is attained at

\[{t_{\min }} = \frac{3\pi }{2} + 2\pi n,\;n \in \mathbb{Z}.\]

The Graph and Properties of the Cosine Function

Graph of the Cosine Function \(y = \cos t\)

When \(t = 0,\) the point \(M\) on the unit circle has coordinates \(M\left( {1,0} \right),\) so

\[\cos \left( {t = 0} \right) = 1.\]

When the point \(M\) is rotated counterclockwise about the origin from \(t = 0\) to \(t = \pi\) the value of cosine decreases from \(1\) to \(-1.\) Then as the angle goes from \(t = \pi\) to \( t= 2\pi,\) the value of \(\cos t\) increases back from \(-1\) to \(1.\) By the cofunction identity,

\[\cos t = \sin \left( {t + \frac{\pi }{2}} \right).\]

This means that we can obtain the graph of cosine function by shifting the sine function by \({\frac{\pi }{2}}\) to the left.

Graphs of sine and cosine functions
Figure 3.

Domain and Codomain of the Cosine Function

Like sine function, the cosine function is defined for all real numbers \(t.\) The codomain (or range) of the cosine function is the interval \(\left[ { - 1,1} \right].\) So, we have

\[\text{dom}\left( {\cos t} \right) = \mathbb{R},\;\text{codom}(\cos t) = \left[ { - 1,1} \right].\]

Parity of the Cosine Function

The cosine function is an even function:

\[\cos \left( { - t} \right) = \cos t.\]

The graph of cosine function is symmetric about the \(y-\)axis.

Periodicity of the Cosine Function

The cosine function is periodic with the least period \(2\pi:\)

\[\cos \left( {t + 2\pi n} \right) = \cos t,\;n \in \mathbb{Z}.\]

Zeros of the Cosine Function

The solutions of the equation \(\cos t = 0\) are given by

\[t = \frac{\pi }{2} + \pi n,\;n \in \mathbb{Z}.\]

Extreme Values of the Cosine Function

The cosine function has maximum value \(\cos t = 1\) at

\[{t_{\max }} = 2\pi n,\;n \in \mathbb{Z}.\]

The minimum value \(\cos t = -1\) occurs at

\[{t_{\min }} = \pi + 2\pi n,\;n \in \mathbb{Z}.\]

See solved problems on Page 2.

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