6.3 – GRAPHING SINE AND COSINE FUNCTIONS

Periodic Function and Period A function is periodic if, for some real

number α, f(x + α) = f(x) for each x in the domain of f.

The least positive value of α for which f(x) = f(x + α) is the period of the function.

Which function is periodic?

Determine the period of the function

Graph of Sine

Look at the value for sin(x) for the domain value between -2π and 2π in multiples of π/4x sin(x)

-2π 0

-7π/4 √2/2

-3π/2 1

-5π/4 √2/2

-π 0

-3π/4 -√2/2

-π/2 -1

-π/4 -√2/2

x sin(x)

0 0

π/4 √2/2

π/2 1

3π/4 √2/2

π 0

5π/4 -√2/2

3π/2 -1

7π/4 -√2/2

2π 0

Graph of Sine

Properties of the Graph of Sine

What is the domain? What is the range? What is the period? Where are all the x-

intercepts? Where is the y-intercept? Where are the

maximums? Where are the minimums?

All real numbers

Set of real numbers between -1 and 1

At y = 1 and occur when x = π/2 + 2πn

at (0, 0)

at πn, where n is any integer

2π

At y = -1 and occur when x = 3π/2 + 2πn

Find each value by referring to the graph of sine

1. sin(5π/2)

2. sin(3π)

Find the values of θ for which sin θ = -1 is true.

Graph y = sinx for the interval of 2π to 4π

Graph of Cosine

Look at the value for cos(x) for the domain value between -2π and 2π in multiples of π/4x cos(x)

-2π 1

-7π/4 √2/2

-3π/2 0

-5π/4 -√2/2

-π -1

-3π/4 -√2/2

-π/2 0

-π/4 √2/2

x cos(x)

0 1

π/4 √2/2

π/2 0

3π/4 -√2/2

π -1

5π/4 -√2/2

3π/2 0

7π/4 √2/2

2π 1

Graph of Cosine

Properties of the Graph of Cosine

What is the domain? What is the range? What is the period? Where are all the x-

intercepts? Where is the y-intercept? Where are the

maximums? Where are the minimums?

All real numbers

Set of real numbers between -1 and 1

At y = 1 and occur when

x = πn, n is an even integer

at (0, 1)

at π/2 + πn, where n is any integer

2π

At y = -1 and occur when x = πn, n is an odd integer

Graphing in your calculator

Make sure you are in radians! Type equation into the y= ‘s menu Go to Zoom Trig before you graph Then your graph will appear accurately

Homework change

Do NOT do #51, change it to #52