Graphs and Properties of Sine and Cosine Functions

Understanding the Graph of the Sine Function and Its Properties

Introduction to the Sine Function

The sine function, denoted as y = sin x, is a fundamental trigonometric function that describes the relationship between an angle and the y-coordinate of a point on the unit circle. In precalculus, we often graph this function using radians as the input variable.

• Ordered pairs on the graph are of the form (x, sin x), where x is measured in radians.

• Key points are determined by evaluating sin x at special angles, such as $0$,$\frac{\pi}{2}$,$\pi$, $\frac{3\pi}{2}$, and $2\pi$.

Example: The point $\left(\frac{\pi}{4}, \sin\frac{\pi}{4}\right) = \left(\frac{\pi}{4}, \frac{\sqrt{2}}{2}\right)$ lies on the graph of y = sin x.

Graph of y = sin x

The graph of y = sin x is a smooth, continuous wave that repeats every $2\pi$ units along the x-axis. This property is called periodicity.

• The function is periodic with period $2\pi$.

• The graph passes through the origin (0, 0).

• Maximum value: 1 at $x = \frac{\pi}{2} + 2\pi n$

• Minimum value: -1 at $x = \frac{3\pi}{2} + 2\pi n$

• Zeros at $x = n\pi$, where n is an integer.

Table: Key Properties of the Sine Function

Property	Value/Description
Domain	$(-\infty, \infty)$
Range	$[-1, 1]$
Period	$2\pi$
y-intercept	0
Zeros	$x = n\pi$
Symmetry	Odd function: $\sin(-x) = -\sin(x)$ (symmetric about the origin)
Maximum	1 at $x = \frac{\pi}{2} + 2\pi n$
Minimum	-1 at $x = \frac{3\pi}{2} + 2\pi n$

Understanding the Graph of the Cosine Function and Its Properties

Introduction to the Cosine Function

The cosine function, denoted as y = cos x, is another fundamental trigonometric function. It describes the relationship between an angle and the x-coordinate of a point on the unit circle.

• Ordered pairs on the graph are of the form (x, cos x), where x is in radians.

• Key points are determined by evaluating cos x at special angles, such as $0$,$\frac{\pi}{2}$,$\pi$, $\frac{3\pi}{2}$, and $2\pi$.

Example: The point $\left(\frac{\pi}{3}, \cos\frac{\pi}{3}\right) = \left(\frac{\pi}{3}, \frac{1}{2}\right)$ lies on the graph of y = cos x.

Graph of y = cos x

The graph of y = cos x is also a smooth, continuous wave, but it starts at its maximum value when x = 0. Like the sine function, it is periodic with period $2\pi$.

• The function is periodic with period $2\pi$.

• The graph passes through (0, 1).

• Maximum value: 1 at $x = 2\pi n$

• Minimum value: -1 at $x = \pi + 2\pi n$

• Zeros at $x = (2n+1)\frac{\pi}{2}$, where n is an integer.

Table: Key Properties of the Cosine Function

Property	Value/Description
Domain	$(-\infty, \infty)$
Range	$[-1, 1]$
Period	$2\pi$
y-intercept	1
Zeros	$x = (2n+1)\frac{\pi}{2}$
Symmetry	Even function: $\cos(-x) = \cos(x)$ (symmetric about the y-axis)
Maximum	1 at $x = 2\pi n$
Minimum	-1 at $x = \pi + 2\pi n$

Quarter Points of Sine and Cosine Functions

Definition and Importance

Quarter points divide one cycle of the sine or cosine curve into four equal parts. These points are useful for sketching accurate graphs.

• For y = sin x: The five quarter points are $(0,0)$, $\left(\frac{\pi}{2},1\right)$, $(\pi,0)$, $\left(\frac{3\pi}{2},-1\right)$, and $(2\pi,0)$.

• For y = cos x: The five quarter points are $(0,1)$, $\left(\frac{\pi}{2},0\right)$, $(\pi,-1)$, $\left(\frac{3\pi}{2},0\right)$, and $(2\pi,1)$.

Amplitude and Period of Sine and Cosine Functions

Amplitude

The amplitude of a sine or cosine curve is half the distance between its maximum and minimum values. For functions of the form $y = A\sin x$ or $y = A\cos x$, the amplitude is $|A|$.

• Range: $[-|A|, |A|]$

Period

The period of a sine or cosine function is the length of one complete cycle. For $y = \sin(Bx)$ or $y = \cos(Bx)$, the period is given by:

$P = \frac{2\pi}{B}$, where $B > 0$

Steps for Sketching Functions of the Form $y = A\sin(Bx)$ and $y = A\cos(Bx)$

1. If $B < 0$, use the even/odd properties of sine and cosine to rewrite the function with $B > 0$.

2. Determine the amplitude and range: Amplitude is $|A|$, range is $[-|A|, |A|]$.

3. Determine the period: $P = \frac{2\pi}{B}$.

4. Divide one period into four equal subintervals to find the quarter points.

5. Multiply the y-coordinates of the standard quarter points by $A$ to get the new y-values.

6. Connect the quarter points to complete the graph.

Determining the Equation from a Graph

Steps for Determining the Equation of $y = A\sin(Bx)$ or $y = A\cos(Bx)$ Given the Graph

1. Decide if the graph is a sine or cosine function (based on whether it passes through the origin or starts at a maximum).

2. Determine the period $P$ from the graph and solve for $B$ using $P = \frac{2\pi}{B}$.

3. Use the amplitude and a given point to solve for $A$.

Additional info: If the graph is shifted or reflected, further analysis is needed to determine phase shift or sign of A.