Trigonometric Functions Chapter 6 Trigonometric Functions

Section 4 Graphs of Sine and Cosine Functions

Definitions: Period  the # at which the graph starts repeating its shape (normally at 2𝜋) Amplitude  the distance from max of the graph to center Center of graph  ½ between max and min values (horizontal line that would divide y-values evenly in half)

Graphing only one period (0 - 2𝜋)  anything outside of this is a repeat Use the unit circle to get the basic points (x, y) 𝜃 = x and sin 𝜃 or cos 𝜃 = y

Understanding the basic y = sin x Basic ordered pairs: at which angle measures do you get values of sin 𝜃 that would be easy to graph? X = 𝜃 y = sin 𝜃 0 0 𝜋/2 1 𝜋 0 3𝜋/2 -1 2𝜋 0

Main Characteristics of sin: Max: 1 Min: -1 Center: 0 “S” shape Starts on center and stops on center + sin  up first - sin  down first Period: 2𝜋 Amplitude: 1 Domain: {x| all real} Range: {y| -1 ≤ y ≤ 1}

Understanding the basic y = cos x Basic ordered pairs: at which angle measures do you get values of cos 𝜃 that would be easy to graph? X = 𝜃 y = cos 𝜃 0 1 𝜋/2 0 𝜋 -1 3𝜋/2 0 2𝜋 1

Main Characteristics of cos: Max: 1 Min: -1 Center: 0 “V” shape + cos  starts and stops at maximum - cos  starts and stops at minimum Period: 2𝜋 Amplitude: 1 Domain: {x| all real} Range: {y| -1 ≤ y ≤ 1}

If y = A sin (wx) + k OR y = A cos (wx) + k Amplitude (A) = |A| Period (T) = 2𝜋/w Center = k

Example: Graph y = sin(x - 𝜋/4) + 2 Basic:

Example: Graph y = -2 cos x Basic:

Find Amplitude and Period Ex Find Amplitude and Period Ex. y = -3 sin(4x) A = T = Y = 2 cos (- 𝜋 2 𝑥)

Example: Find the equation of the sin function with the following description: Amplitude = 3 and period = 𝜋

Example: (#73 on page 408) Max: 3 Min: -3 Center: 0 Starts at: min  Amp: 3 Per: 4𝜋  w =

Example: (#75 on page 408) Max: 3/4 Min: -3/4 Center: 0 Starts at: center  Amp: 3/4 Per: 1 w =

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