Graphs of the Sine and Cosine Functions

Graphing Trigonometric Functions Graph in xy-plane Write functions as y = f(x) = sin x y = f(x) = cos x y = f(x) = tan x y = f(x) = csc x y = f(x) = sec x y = f(x) = cot x Variable x is an angle, measured in radians Can be any real number

Graphing the Sine Function Periodicity: Only need to graph on interval [0, 2¼] (One cycle) Plot points and graph

Properties of the Sine Function Domain: All real numbers Range: [{1, 1] Odd function Periodic, period 2¼ x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, … y-intercept: 0 Maximum value: y = 1, occurring at Minimum value: y = {1, occurring at

Transformations of the Graph of the Sine Functions Example. Problem: Use the graph of y = sin x to graph Answer:

Graphing the Cosine Function Periodicity: Again, only need to graph on interval [0, 2¼] (One cycle) Plot points and graph

Properties of the Cosine Function Domain: All real numbers Range: [{1, 1] Even function Periodic, period 2¼ x-intercepts: y-intercept: 1 Maximum value: y = 1, occurring at x = …, {2¼, 0, 2¼, 4¼, 6¼, … Minimum value: y = {1, occurring at x = …, {¼, ¼, 3¼, 5¼, …

Transformations of the Graph of the Cosine Functions Example. Problem: Use the graph of y = cos x to graph Answer:

Sinusoidal Graphs Graphs of sine and cosine functions appear to be translations of each other Graphs are called sinusoidal Conjecture.

Amplitude and Period of Sinusoidal Functions Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAj Number jAj is the amplitude

Amplitude and Period of Sinusoidal Functions Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAj Number jAj is the amplitude

Amplitude and Period of Sinusoidal Functions Period of y = sin(!x) and y = cos(!x) is

Amplitude and Period of Sinusoidal Functions Cycle: One period of y = sin(!x) or y = cos(!x)

Amplitude and Period of Sinusoidal Functions Cycle: One period of y = sin(!x) or y = cos(!x)

Amplitude and Period of Sinusoidal Functions Theorem. If ! > 0, the amplitude and period of y = Asin(!x) and y = Acos(! x) are given by Amplitude = j Aj Period = .

Amplitude and Period of Sinusoidal Functions Example. Problem: Determine the amplitude and period of y = {2cos(¼x) Answer:

Graphing Sinusoidal Functions One cycle contains four important subintervals For y = sin x and y = cos x these are Gives five key points on graph

Graphing Sinusoidal Functions Example. Problem: Graph y = {3cos(2x) Answer:

Finding Equations for Sinusoidal Graphs Example. Problem: Find an equation for the graph. Answer:

Key Points Graphing Trigonometric Functions Graphing the Sine Function Properties of the Sine Function Transformations of the Graph of the Sine Functions Graphing the Cosine Function Properties of the Cosine Function Transformations of the Graph of the Cosine Functions

Key Points (cont.) Sinusoidal Graphs Amplitude and Period of Sinusoidal Functions Graphing Sinusoidal Functions Finding Equations for Sinusoidal Graphs

Graphs of the Tangent, Cotangent, Cosecant and Secant Functions Section 5.5

Graphing the Tangent Function Periodicity: Only need to graph on interval [0, ¼] Plot points and graph

Properties of the Tangent Function Domain: All real numbers, except odd multiples of Range: All real numbers Odd function Periodic, period ¼ x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, … y-intercept: 0 Asymptotes occur at

Transformations of the Graph of the Tangent Functions Example. Problem: Use the graph of y = tan x to graph Answer:

Graphing the Cotangent Function Periodicity: Only need to graph on interval [0, ¼]

Graphing the Cosecant and Secant Functions Use reciprocal identities Graph of y = csc x

Graphing the Cosecant and Secant Functions Use reciprocal identities Graph of y = sec x

Key Points Graphing the Tangent Function Properties of the Tangent Function Transformations of the Graph of the Tangent Functions Graphing the Cotangent Function Graphing the Cosecant and Secant Functions

Phase Shifts; Sinusoidal Curve Fitting

Graphing Sinusoidal Functions y = A sin(!x), ! > 0 Amplitude jAj Period y = A sin(!x { Á) Phase shift Phase shift indicates amount of shift To right if Á > 0 To left if Á < 0

Graphing Sinusoidal Functions Graphing y = A sin(!x { Á) or y = A cos(!x { Á): Determine amplitude jAj Determine period Determine starting point of one cycle: Determine ending point of one cycle:

Graphing Sinusoidal Functions Graphing y = A sin(!x { Á) or y = A cos(!x { Á): Divide interval into four subintervals, each with length Use endpoints of subintervals to find the five key points Fill in one cycle

Graphing Sinusoidal Functions Graphing y = A sin(!x { Á) or y = A cos(!x { Á): Extend the graph in each direction to make it complete

Graphing Sinusoidal Functions Example. For the equation (a) Problem: Find the amplitude Answer: (b) Problem: Find the period (c) Problem: Find the phase shift

Finding a Sinusoidal Function from Data Example. An experiment in a wind tunnel generates cyclic waves. The following data is collected for 52 seconds. Let v represent the wind speed in feet per second and let x represent the time in seconds. Time (in seconds), x Wind speed (in feet per second), v 21 12 42 26 67 41 40 52 20

Finding a Sinusoidal Function from Data Example. (cont.) Problem: Write a sine equation that represents the data Answer:

Key Points Graphing Sinusoidal Functions Finding a Sinusoidal Function from Data