6.1 Graphs of the Sine and Cosine Functions - Precalculus 2e | OpenStax

Learning Objectives

• Graph variations of sine and cosine functions.

• Use phase shifts of sine and cosine curves.

Light and Wave Properties

• Light can be separated into colors due to its wavelike properties.

• White light is composed of all the colors of the rainbow in wave form.

• Individual colors appear when white light passes through an optical prism, which separates the waves according to their wavelengths.

• Light waves can be represented using trigonometric functions, specifically sine functions.

Graphing Sine and Cosine Functions

Sine Function

• The sine function relates real number values to the coordinates of points on the unit circle.

• A table of values can be used to sketch its graph.

• Values of sine are positive in quadrants I and II, and negative in quadrants III and IV.

  • Table of values (sine function):

  • [ \text{x:} , 0, , \frac{\pi}{6}, , \frac{\pi}{4}, , \frac{\pi}{3}, , \frac{\pi}{2}, , \pi, , \frac{3\pi}{2}, , 2\pi} ]

  • [ \text{sin(x):} , 0, , \frac{1}{2}, , \frac{\sqrt{2}}{2}, , \frac{\sqrt{3}}{2}, , 1, , 0, , -1, , 0} ]

Cosine Function

• The cosine function is similarly evaluated using a table of values.

• The cosine values create a graph similar to sine but shifted horizontally.

  • Table of values (cosine function):

  • [ \text{x:} , 0, , \frac{\pi}{6}, , \frac{\pi}{4}, , \frac{\pi}{3}, , \frac{\pi}{2}, , \pi, , \frac{3\pi}{2}, , 2\pi} ]

  • [ \text{cos(x):} , 1, , \frac{\sqrt{3}}{2}, , \frac{\sqrt{2}}{2}, , \frac{1}{2}, , 0, , -1, , 0, , -1} ]

Properties of Sine and Cosine Functions

• Both functions are defined for all real numbers, and their ranges are ([-1, 1]).

• Both are periodic functions with a period of (2\pi).

  • A function is periodic if it repeats its values in regular intervals.

• Symmetries:

  • Sine function is symmetric about the origin (odd function): (sin(-x) = -sin(x)).

  • Cosine function is symmetric about the y-axis (even function): (cos(-x) = cos(x)).

Sinusoidal Functions

• Sinusoidal functions are variations of sine and cosine functions with potential characteristics: amplitude, period, phase shift, and vertical shift.

• Amplitude: The height of a wave from its midline, given by |A| in the equation (y = A sin(Bx) + D).

• Period: The length of one complete cycle of the wave, calculated as (P = \frac{2\pi}{|B|}).

  • A larger |B| compresses the wave; a smaller |B| stretches the wave.

Phase Shift and Vertical Shift

• Phase Shift ( (C) ): The horizontal shift of the function.

  • Positive C shifts to the right, negative C shifts to the left.

• Vertical Shift ( (D) ): The upward/downward shift of the graph.

  • Any value of D other than zero shifts the graph up or down.

Examples and Practice Problems

• Example 1: Identify the period of a sine or cosine function from its formula.

• Example 2: Determining the amplitude of a sinusoidal function.

• Several practice problems for graphing, determining properties, and translating sinusoidal functions are provided.

Real-World Applications

• Sinusoidal functions can model various real-world phenomena, such as:

  • Ocean waves

  • Circular motion (e.g., Ferris wheel height)

  • Oscillations of springs or weights.