# Precalculus

## Trigonometry # 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.

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.

### 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.

### 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.