Trigonometric Functions: Graphs of Sine and Cosine Functions, Lecture notes of Advanced Calculus

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Chapter 6

Trigonometric Functions

Section 4

Graphs of Sine and Cosine Functions

Graphing only one period (0 - 2 𝜋)  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 3 𝜋/2 - 1 2 𝜋 0

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 3 𝜋/2 0 2 𝜋 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}

Example: Graph y = sin(x - 𝜋/4) + 2 Basic: (0, 0) (pi/2, 1) (pi, 0) (3pi/2, - 1) (2pi, 0)

• pi/4  right pi/ (pi/4, 0) (3pi/4, 1) (5pi/4, 0) (7pi/4, - 1) (9pi/4, 0) +2  up 2 (pi/4, 2) (3pi/4, 3) (5pi/4, 2) (7pi/4, 1 ) (9pi/4, 2)

Example: Graph y = - 2 cos x Basic: (0, 1) (pi/2, 0) (pi, - 1) (3pi/2, 0) (2pi, 1)

• 2  stretch/reflection (0, - 2) (pi/2, 0) (pi, 2) (3pi/2, 0) (2pi, - 2)

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  - cos Amp: 3 Per: 4𝜋  w = 1/

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