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, 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 $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$.

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 is a smooth, continuous wave that repeats every $2\pi$ units.

• It passes through the origin (0, 0), reaches a maximum of 1 at $x = \frac{\pi}{2}$, a minimum of -1 at $x = \frac{3\pi}{2}$, and returns to 0 at $x = 2\pi$.

Characteristics of the Sine Function

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

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

• y-intercept: 0

• x-intercepts: $x = n\pi$, where n is an integer.

• Odd function: $\sin(-x) = -\sin(x)$ (symmetric about the origin).

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

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

Understanding the Graph of the Cosine Function and Its Properties

Introduction to the Cosine Function

The cosine function, 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 are of the form (x, cos x), with x in radians.

• Key points are found at $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$.

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 is a smooth, continuous wave that repeats every $2\pi$ units.

• It starts at (0, 1), reaches 0 at $x = \frac{\pi}{2}$, -1 at $x = \pi$, 0 at $x = \frac{3\pi}{2}$, and returns to 1 at $x = 2\pi$.

Characteristics of the Cosine Function

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

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

• y-intercept: 1

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

• Even function: $\cos(-x) = \cos(x)$ (symmetric about the y-axis).

• Maximum value: 1 at $x = 2n\pi$

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

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.

Function	Quarter Points (x, y)
y = sin x	(0, 0), $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, $(2\pi, 0)$
y = cos x	(0, 1), $(\frac{\pi}{2}, 0)$, $(\pi, -1)$, $(\frac{3\pi}{2}, 0)$, $(2\pi, 1)$

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 is the length of one complete cycle of the function. For $y = \sin(Bx)$ or $y = \cos(Bx)$, the period is:

$P = \frac{2\pi}{B}$, 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 smoothly to complete one cycle of 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/minimum).

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

3. Use the amplitude and points on the graph to solve for $A$.

Additional info: If the graph is shifted horizontally or vertically, further analysis is needed to determine phase shift and vertical translation, which are covered in more advanced sections.