4.4 Graphing sin and cos
Functions
5–Minute Check 1
Let (–5, 12) be a point on the terminal side of an
angle θ in standard position. Find the exact
values of the six trigonometric functions of θ.
Let
, where cos θ < 0. Find the exact
values of the five remaining trigonometric
functions of θ.
Let (–5, 12) be a point on the terminal side of an
angle θ in standard position. Find the exact
values of the six trigonometric functions of θ.
Let
, where cos θ < 0. Find the exact
values of the five remaining trigonometric
functions of θ.
• sinusoid
• amplitude
• frequency
• phase shift
• vertical shift
• midline
Key Concept 1
Sine Function
Rad
y
0
𝜋
6
𝜋
2
5𝜋
6
𝜋
7𝜋
6
3𝜋
2
11𝜋
6
2𝜋
Equations of Sine and Cosine
𝑦 = 𝑎 sin 𝑏𝑥 + 𝑐 + 𝑑
𝑦 = 𝑎 cos 𝑏𝑥 + 𝑐 +𝑑
Amplitude : 𝑎
2𝜋
Period : 𝑏
1
Increments : 4 𝑝
Graph Vertical Dilations of Sinusoidal
Functions
Describe how the graphs of f(x) = sin x and
g(x) = 2 sin x are related. Then find the amplitude of g(x), and sketch two
periods of both functions on the same coordinate axes.
X-intercepts:
Maximum :
Period:
Minimum:
Describe how the graphs of f(x) = sin x and
𝟏
g(x) = 𝟒sin x are related. Then find the amplitude of g(x), and sketch two
periods of both functions on the same coordinate axes.
𝑦=
1
sin 𝑥
4
X-intercepts:
Maximum :
Minimum:
Period:
Cosine function
Rad
x
0
𝜋
3
𝜋
2
2𝜋
3
𝜋
4𝜋
3
3𝜋
2
5𝜋
3
2𝜋
Equations of Sine and Cosine
𝑦 = 𝑎 sin 𝑏𝑥 + 𝑐 + 𝑑
𝑦 = 𝑎 cos 𝑏𝑥 + 𝑐 +𝑑
Amplitude : 𝑎
2𝜋
Period : 𝑏
1
Increments : 4 𝑝
Describe how the graphs of f(x) = cos x and
g(x) = 5 cos x are related.
Maximum :
X-intercepts:
Minimum:
Period:
Describe how the graphs of f(x) = cos x and
g(x) = –6 cos x are related.
Maximum :
Minimum:
X-intercepts:
Period:
Key Concept 3
Graph Horizontal Dilations of Sinusoidal
Functions
Describe how the graphs of f(x) = cos x and
g(x) = cos
are related. Then find the period of
g(x), and sketch at least one period of both functions on the same coordinate
axes.
Extrema:
Intercepts:
Increments:
Period:
Graph Horizontal Dilations of Sinusoidal
Functions
Sketch the curve through the indicated points for each function, continuing the
patterns to complete one full cycle of each.
Describe how the graphs of f(x) = sin x and
g(x) = sin 4x are related.
Extrema:
Intercepts:
Increments:
Period:
Homework
• Complete the worksheet
Warm Up
1. Find all values and graph
2. 𝑦 = 4𝑐𝑜𝑠3𝜃
3. 𝑦 =
1
𝜃
sin
2
2
1.
𝑦 = 4𝑐𝑜𝑠3𝜃
1
𝜃
𝑦 = sin
2
2
Homework answers
Key Concept 5
Steps for graphing Sin and Cos
• Identify a, b, c, d, period and increments and ps.
• Write your phase shift, increment and “c” with a common
denominator to make the math easier
• Create your table
– starting at the ps
– use your increments.
𝜋
– Ex. If increments are and your ps =0 then x or 𝜃 𝑣𝑎𝑙𝑢𝑒𝑠 are:
4
𝜋 2𝜋 3𝜋 4𝜋
0, , , ,
•
•
•
•
4
4
4
4
Plug in the 𝜃 values and do the math
Set up your graph using your 𝜃 values
Graph
Identify the midline
5𝜋
2 cos 𝜃 −
4
Amp:
b:
c:
Per:
Increments:
PS:
2𝜋
𝑦 = 3 sin 𝜃 +
3
Amp:
b:
c:
Per:
Increments:
PS:
1
𝜃 7𝜋
𝑦 = sin
−
+2
2
4
6
Amp:
b:
c:
Per:
Increments:
PS:
𝑦 = −3 − 3 cos 3𝜃
Amp:
b:
c:
Per:
Increments:
PS:
𝜋
𝑦 = 2 sin 4𝜃 +
+2
3
Amp:
b:
c:
Per:
Increments:
PS:
Key Concept 4
State the amplitude, period, frequency, phase shift, and vertical shift of
.
Use Frequency to Write a
Sinusoidal Function
MUSIC A bass tuba can hit a note with a frequency of 50 cycles per second
(50 hertz) and an amplitude of 0.75. Write an equation for a cosine function
that can be used to model the initial behavior of the sound wave associated
with the note.
The general form of the equation will be y = a cos bt,
where t is the time in seconds. Because the amplitude
is 0.75, |a| = 0.75. This means that a = ±0.75.
The period is the reciprocal of the frequency or
Use this value to find b.
.
Use Frequency to Write a
Sinusoidal Function
Period formula
period =
|b| = 2π(50) or 100π
Solve for |b|.
Solve for b.
By arbitrarily choosing the positive values of a and b,
one cosine function that models the initial behavior is
y = 0.75 cos 100πt.
Answer:
Sample answer: y = 0.75 cos 100πt
MUSIC In the equal tempered scale, F sharp has a frequency of 740 hertz.
Write an equation for a sine function that can be used to model the initial
behavior of the sound wave associated with
F sharp having an amplitude of 0.2.
Key Concept 7