Download 1-7-15 notes .notebook

1­7­15 notes .notebook
January 07, 2015
Welcome Back and
Introduction to Trig Graphs
HAPPY 2015!!!
Types of Trig Graphs:
First Journal Entry of 2015:
1. Sine Graphs/Inverse Sine Graphs
Write down your new year's resolution!
2. Cosine Graphs/Inverse Cosine Graphs
3. Tangent Graphs/Inverse Tangent Graphs
Dec 18­8:17 AM
0º
Dec 18­8:26 AM
Trig Graphs
Sine Graphs
(Recall the chart below from before break)
To get the values we will need to graph a sine curve, we will simply need to focus on the exact values for sine on our exact value chart.
120º 135 150 180 210 225º 240º 270º 300 315 330º
360º
º
º
º
º
º
º
30º 45º 60º 90º
0º
sin
(θ)
cos
(θ)
30º
45º 60º
90º
120º 135º 150º 180º 210º 225º 240º 270º 300º 315º 330º 360º
sin(θ)
Dec 18­8:29 AM
Dec 18­8:33 AM
Vocabulary
Graphing the Sine Curve
There are first some things you need to know:
About the axis:
y
3
1. y­axis: The scale of the y­
axis is numbers (i.e. 0, 1, 2, 3, etc...)
2
1
x
­90º ­60º ­45º ­30º
­1
30º 45º 60º 90º
­ This means that the exact values we got from our chart will be the y­
values.
2. x­axis: The scale of the x­
axis is degrees (0º, 30º, 45º, 60º, etc...)
­2
­3
Amplitude: ­ The amplitude of the trigonometric functions is one­half the positive distance between the maximum and minimum values of the function. Or the height of the function.
Period:
­ One complete repetition of a function is called a cycle. The period of the function is the horizontal length of one complete cycle. Or the length of one cycle.
Frequency:
­ The frequency of the function is the number of cycles it completes in a given interval. This interval is generally 360º for sine and cosine functions.
Domain:
­ The domain is the set of all first elements of ordered pairs (x­coordinates). Range:
­ The range is the set of all second elements of ordered pairs (y­coordinates).
­ This means that the degree values from the top of our chart will be the x­values.
Dec 18­8:37 AM
Jan 5­8:24 AM
1
1­7­15 notes .notebook
January 07, 2015
y
1
Using your calculator, please fill in the decimal approximations for the following chart:
.5
x
30º 60º 120º 150º 210º 240º
Means my
300º 330º
45º 90º 135º 180º 225º 270º 315º 360º
Means my first point second point is: (0º,0)
is: (30º,.5)
­.5
­1
sin(θ)
0º
30º
0
.5
45º
.707
60º
.866
Jan 5­8:13 AM
90º
1
120º
.866
135º 150º
.707
.5
180º 210º 225º
0
­.5
­ .707
240º
270º
­.866 ­1
300º 315º
­.866 ­.707
330º
­.5
360º
0
Dec 18­8:53 AM
Ms. R, how does this relate to the unit circle?
y
1
.5
x
30º 60º 120º 150º 210º 240º 300º 330º
y = sin(x) [basic sine curve]
1. amplitude (height) = 1
45º 90º 135º 180º 225º 270º 315º 360º
­.5
­1
2. period (length of
one cycle) = 360º
3. Frequency (how many cycles per 1 period) = 1
4. Domain: { x | x ∈ R}
5. Range: { y | ­1 ≤ y ≤ 1}
Dec 18­9:25 AM
Dec 18­11:22 AM
Sine Graphs ­ Day 2
Complete the chart below and graph y=sin(x) in the interval 0 ≤ x ≤ 360 :
Where can we see sine graphs in "real life"?
s
ave
Tida
w
d
oun
.5
l wa
S
1
ve g
raph
s
30º ­ .5
res
60º 120º 150º 210º 240º 300º 330º
45º 90º 135º 180º 225º 270º 315º 360º
­1
eratu
Temp
Jan 5­10:40 AM
Jan 5­10:52 AM
2
1­7­15 notes .notebook
January 07, 2015
On your second piece of graph paper:
What if I asked you to graph
Graph y=sin(x) in the interval 0 ≤ x ≤ 720
y=sin(x) in the interval 0 ≤ x ≤ 720?
1
Writing down ALL of those degree measures on the x­axis would
be extremely tedious, don't you think?
.5
It would be ok for you to only plot points for our
quadrantal angle measures.
90º
What were those measures again?
­ .5
0º, 90º, 180º, 270º, 360º
­1
180º
270º
360º 450º
540º
630º
720º
WHY ARE THESE IMPORTANT?
Jan 6­8:24 AM
Jan 6­8:31 AM
General form of the Sine Function
y = A sin(B(x ­ C)) + D
Example: Find the amplitude,
frequency, and period of the
Amplitude: |A|
Find the amplitude, frequency, and period of the
following functions:
1. y = 7sin(4x)
3. y = 5sin(5x)
2. y = ­3sin(2x)
4. y = ­8sin(9x)
following function.
Frequency: B
y = 3sin(2x)
Period: 360
Amplitude:
B
Frequency:
Vertical Shift: D
Period:
Horizontal Shift: C
Jan 6­8:42 AM
Jan 6­8:50 AM
Amplitude
Formula: |maximum­minimum|
2
2.
1.
3.
Jan 6­8:36 AM
Jan 7­2:13 PM
3
1­7­15 notes .notebook
January 07, 2015
Write the equation for the following graph:
Frequency
(How many times you see the complete sine curve
between 0º and 360º)
1
1
.5
.5
­ .5
90º 180º 270º 360º
­ .5
2
1
90º 180º 270º 360º
­1
­1
Steps:
1. Write out the
general formula for
sine.
­1
90º
180º
270º
360º
­2
2. Find the
amplitude,
frequency, and see
if there are any
vertical or
horizontal shifts.
3. Fill in the general
formula with your
findings.
Jan 6­8:42 AM
Jan 6­9:08 AM
Write the equation for the following graph:
3
2
1
­1
­2
­3
90º
180º
270º
Jan 6­9:11 AM
360º
Jan 6­9:20 AM
4