Graphing Trigonometry - St. Francis Preparatory School



Graphing Trigonometry

y = sinx

|X |0º |90º |180º |270º |360º |

|Y |0 |1 |0 |-1 |0 |

(These values came directly from the quadrantal angles found on the unit circle)

We are now going to graph this curve on Graph Paper.

Guidelines for graphing the sine curve:

1. We are going to hold the graph paper sideways

2. For this example we only need the first quadrant. (You can tell by where the table

starts. Here we started at zero.)

3. If you notice, the numbers on the table are very large. They go up to 360º and could go

further. Because of this we are ALL going to use the same intervals for EVERY graph no matter the table.

For the X- AXIS : Starting at 0, every six boxes is 90º. (Each box is 15º)

For the Y-AXIS : Starting at 0, every 4 boxes is 1. (Each box is .25)

Do Now: Graph y = sinx from the table above. ON GRAPH PAPER.

(It should look something like this)

• Make sure to label everything*

The sine curve is a continuous curve that passes through (0,0)

y = cosx

|X |0º |90º |180º |270º |360º |

|Y |1 |0 |-1 |0 |1 |

(These values came directly from the quadrantal angles found on the unit circle)

The cosine curve is a continuous curve passing through (0,1)

Do Now: Graph y = cosx from the table above ON GRAPH PAPER.

(It should look something like this)

General Equations:

y = asinbx

y = acosbx

Notice a is the coefficient of sine or cosine and b is the coefficient of x.

a: amplitude = |a|

- The amplitude tells you how high and low the curve will go.

- It gives the range of the graph and can be stated as : -a ≤ y ≤ a

- It gives the min and max value points (state as a positive number)

b: frequency |b|

- The frequency tells you how many complete curves you will have within 360º.

Complete sine curve Complete cosine curve

Period (p): [pic]

- Tells you how many degrees it takes to draw each curve.

Example: State the amplitude, period, frequency and range of each function.

1. y = 3 sin 2x 2. y = cos(-4)x

a = 3 a = 1

b = 2 b = 4

p = [pic]= 180º p = [pic]= 90º

range: -3 ≤ y ≤ 3 range: -1≤ y ≤ 1

Do Now: Find the amplitude, period, frequency and range of each function.

1. y = [pic]sin x 2. y = -2 sin [pic]x 3. y = 2 cos 3x

4. y = [pic]cos 2x 5. y = -4 sin 12x 5. y = - cos (-3)x

Graphing ANY sine or cosine curve:

Example: Graph y = sin2x on the interval 0≤ x ≤ 360º

1. Start by labeling the amplitude, period, frequency and range

2. You must create your table BUT you have to know what angle value to count by. (It

will not always be 90º) To tell what you will count by, look at b.

If b ≤ 1 count by 90ºs

If b = 2 count by 45ºs

If b = 3 count by 30ºs

(You will only have to know the 3 b’s above)

3. Also remember if the interval is given in radians, you must put a row in your table of

all the radian values.

Let’s now create a table for the example above: y = sin2x

X |0º |45º |90º |135º |180º |225º |270º |315º |360º | |Y |0 |1 |0 |-1 |0 |1 |0 |-1 |0 | |* Notice b = 2 so we counted by 45º *

Do Now: Graph the above table on GRAPH PAPER. LABEL!

Graphing Trig Functions on the Calculator

1. Put the equation into y = (Remember fractions need parentheses)

2. Hit Zoom 7 for the trig graph (Notice that is shows you from -360º to 360º so you may

only be graphing half of it on your graph paper)

3. To see the table with the values you want you must hit 2nd Window (Tbl set)

You want Tbl Start = 0 (Because your chart starts at 0)

You must change Δ Tbl = 30, 45, or 90 (Depending on what you are counting by)

4. Now you can hit 2nd Graph and all of your table values should be there.

Example: Graph y = [pic]cos 3x on the interval –π ≤ x ≤ π

(Notice here that the interval is in radians. Therefore we must put an

additional radian row into our table. You can convert to degrees first to see

what numbers we are working with)

a = ½ b = 3 p = 120º range: - ½ ≤ y ≤ ½

X |-180º |-150º |-120º |-90º |-60º |-30º |0º |30º |60º |90º |120º |150º |180º | |Radians |- π |[pic] |[pic] |[pic] |[pic] |[pic] |0 |[pic] |[pic] |[pic] |[pic] |[pic] |π | |Y |- ½ |0 |½ |0 |- ½ |0 |½ |0 |- ½ |0 |½ |0 |- ½ | |

****Important: When the interval is given in radians you must also put the radians on the graph.****

Do Now: Graph the above table ON GRAPH PAPER.

Practice: Graph each function below. Create a table and graph the function on graph paper.

1. y = 2 cos x

2. y = sin 2x

3. y = 3 cos 2x

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