Algebra 2 Notes
Section 14.0: Graphing Basic Trigonometric Functions
Objective(s):
Vocabulary (see page 908):
I. Amplitude:
II. Periodic Function
III. Cycle:
IV. Period:
Use the unit circle to complete the table of values:

θ


2
0
2
3
2
π
2π
y = sin θ
y = cos θ
Using the table of values, graph both sine and cosine functions.
y = sin x

π
2
y = cos x
1
1
.5
.5
0
π
2
π
3π
2
2π

π
2
0
-.5
-.5
-1
-1
π
2
π
3π
2
2π
Identify the key elements for each function:
Maximum
y = sin θ
y = cos θ
Minimum
Amplitude
Period
Domain
Range
Notes 14.0 page 2
Complete the table of values:

Θ


2


4
0
4

2
3
4
π
5
4
3
2
y = tan θ
Using the table of values, graph the tangent function.
y = tan x
4
2

π
2
0
π
2
π
3π
2
2π
-2
-4
dentify the key elements for the tangent function:
Maximum
y = tan θ
Minimum
Amplitude
Period
Domain
Range
Algebra 2 Notes
Section 14.1: Graph Trigonometric Functions with Scale Changes
Objective(s):
Vocabulary:
I. Frequency:
Amplitude and Period for y = a sin bx and y = a cos bx
Amplitude:
Period:
Graphing Key Points
These are the x-values where
Examples:
Graph each of the following functions using key points.
1. y = sin 3x

2.
π
6
2
.5
1
0
π
6
π
3
π
2
2π
3

0
-1
-1
-2
4.
π
6
π
2
-.5
3. y = cos 3x

y = 2 sin x
1
8
.5
4
π
6
π
3
π
2
2π
3
π
π
2
π
3π
2
2π
3π
2
2π
y = 5 cos x
1
0
π
2

π
2
0
-.5
-4
-1
-8
Notes 14.1 page 2
5. y = 3 sin 2x

6.
y = 3 cos 2x
4
4
2
2
0
π
2
π
2
π
3π
2
2π

π
2
-2
-2
-4
-4
7. y = 3 tan x
8.
4
π
2
0
π
2
π
π
2
π
3π
2
2π
3π
2
2π
1 
y  2 tan  x 
2 
4
2

0
2
π
2
π
3π
2
2π

π
2
0
-2
-2
-4
-4
Algebra 2 Notes
Section 14.2: Translate and Reflect Trigonometric Graphs
Objective(s):
Vocabulary:
I. Translation (p. 123):
II. Reflection (p. 1071):
III. Amplitude (p. 908):
IV. Period (p. 908):
Translations of Sine and Cosine Graphs:
y = a sin b(x – h) + k
y = a cos b(x – h) + k
Step 1:
Step 2:
Step 3:
Step 4:
Examples:
Graph each of the following functions using key points.
1.
y
1
cos 2x  2
2

π
2
2.
1
y  2sin (x  π)
2
2
2
1
1
0
π
2
π
3π
2
2π
 2π
0
-1
-1
-2
-2
Reflections:
When a is negative, reflect the graph across the midline (x-axis or "fake x-axis") of the graph.
2π
4π
6π
8π
Notes 14.2 page 2
3.
π

y   cos  x  
2


π
2
4.
y   3 sin
1
x2
2
4
8
2
4
0
π
2
π
3π
2
π
2π
π
0
-2
-4
-4
-8
2π
3π
4π
Translations of the Tangent Graph:
y = a tan b(x – h) + k
Step 1:
Step 2:
Step 3:
Step 4:
5.
π

y  2 tan  x  
2

6.
π

y  3 tan  x  
4

4
2
2

1
π
2
0
-2

π
2
0
-1
-2
π
2
π
3π
2
2π
-4
π
2
π
3π
2
2π
Algebra 2 Notes
Section 14.3: Verify Trigonometric Identities
Objective(s):
Vocabulary:
I. Trigonometric Identity:
Fundamental Trigonometric Identities:
Reciprocal Identities
csc θ =
sec θ =
cot θ =
Tangent and Cotangent Identities
tan θ =
cot θ =
Pythagorean Identities
Cofunction Identities
Negative Angle Identities
Examples:
1.
Given cos   
4
π
 θ  π , find the values of the other five trigonometric functions of θ.
and
5
2
Verifying Identities (p. 926):
When verifying an identity,
Notes 14.3 page 2
2.
a)
Simplify the expression.
tan (-θ) cos θ
b) sin θ + cos θ cot θ
3.
a)
Verify the identity.
sec θ – cos θ = sin θ tan θ
b) sin θ (tan θ + cot θ) = sec θ
2
2
2
c) csc x(1 – sin x) = cot x
d) cos x csc x tan x = 1
Algebra 2 Notes
Section 14.4: Solve Trigonometric Equations
Objective(s):
Vocabulary:
I. Extraneous solution (p. 52):
II. General solution (margin on p.931):
Examples:
Solve the equation in the interval 0 < x < 2π.
1. 2cos x + 1 = 0
2.
3 – tan x = 0
Find the general solutions.
3. 2 sin x + 4 = 5
4.
2cos x - cos x = 0
6.
1 - cos x =
5.
3
sin x - sin x = 0
2
3
3 sin x
Notes 14.4 page 2
Solve the equation in the interval 0 < x < π.
7. 2 sin x = csc x
Solve the equation in the interval 0 < x < 2π.
9. sin x - cos x = 1
8.
2
2
tan x - sin x ∙tan x = 0