4.5 Graphs of Sine and Cosine Functions
What are we learning?
-Sketching the graphs of sine and cosine.
-Using amplitude and period to sketch.
-Sketching translations.
Why are we learning it?
Sine and cosine functions are often used in scientific
calculations.
Fill in the table. Then use the points to sketch a graph of y = sinx.
X
-2π
-3π/2
-π
-π/2
0
π/4
π/2
3π/4
π
5π/4
3π/2
7π/4
2π
π
5π/4
3π/2
7π/4
2π
y=
sinx
What is the domain of sinx?
What is the range?
Fill in the table. Then use the points to sketch a graph of y = cosx
X
-2π
-3π/2
-π
-π/2
0
π/4
π/2
3π/4
y=
sinx
What is the domain of cosx?
Amplitude-
Period-
What is the range?
Exploration: Use a calculator to sketch the functions. Identify the period and amplitude. Answer the questions.
a) y = 2sinx
period:
amplitude:
b) y = 3cosx
period:
amplitude:
c) y = (1/2)sinx
period:
amplitude:
d) y = -4sinx
period:
amplitude:
What effect does the constant have on these graphs compared to the graph of sinx or cosx?
In general, what can you say about the graph of y = asinx or y = acosx where a is any constant?
e) y = sin2x
period:
amplitude:
f) y = cos3x
period:
amplitude:
g) y =sin(x/2)
period:
amplitude:
What effect does the constant have on these graphs compared to the graph of sinx or cosx?
In general, what can you say about the graph of y=sinbx or y=cosbx where b is any constant?
h) y = sinx +2
period:
amplitude:
i) y = cosx – 4
period:
amplitude:
j) y = 3 - sinx
period:
amplitude:
What effect does the constant have on these graphs compared to the graph of sinx or cosx?
In general, what can you say about the graph of y=sinx + d or y=cosx + d where d is any constant?
k) y = sin(x +π)
period:
amplitude:
l) y = cos(x – π/4)
period:
amplitude:
What effect does the constant have on these graphs compared to the graph of sinx or cosx?
In general, what can you say about the graph of y=sin(x + c) or y=cos(x + c) where c is any constant?
For y = asin(bx – c) + d
and y = acos(bx – c) + d:
a:
b:
c:
d:
Example 1: Sketch the graph of y = 3cosx.
Example 2: Sketch the graph of y = (1/3)sin2x.
Example 3: Sketch the graph of y = 2sinx -1.
Example 4: Sketch the graph of y = 3 – sin(x/2)
4.6 Graphs of Other Trigonometric Functions
What are we learning?
-Graphing the other trigonometric functions.
-Identifying vertical asymptotes.
Tangent Function
Where is the tangent function undefined?
How can we write all of the points at once?
What happens to the graph of tangent at these points?
What is the domain of the tangent function?
What is the range of the tangent function?
Cotangent Function
Where is the cotangent function undefined?
How can we write all of the points at once?
What happens to the graph of cotangent at these points?
What is the domain of the cotangent function?
What is the range of the cotangent function?
Why are we learning it?
Trigonometric functions can be used in many real world
applications.
Secant Function
Where is the secant function undefined?
How can we write all of the points at once?
What happens to the graph of secant at these points?
What is the domain of the secant function?
What is the range of the secant function?
Cosecant Function
Where is the cosecant function undefined?
How can we write all of the points at once?
What happens to the graph of cosecant at these points?
What is the domain of the cosecant function?
What is the range of the cosecant function?
4.7 Inverse Trigonometric Functions
What are we learning?
-Evaluating inverse trigonometric functions.
-Sketching inverse sine and cosine.
Why are we learning it?
Inverse trigonometric functions are used to solve real
world applications.
What must be true about a function for it to have an inverse function?
Do the trigonometric functions have that quality?
Inverse Sine Function
Example 1: If possible, find the exact value.
b) sin-1(1/2)
a) arcsin(-1)
c) arcsin(√3)
Example 2: Sketching arcsine
Fill in the table, and then use the points to sketch a graph of arcsine.
x
Arcsin(x)
-1
−
√2
2
-1/2
0
1/2
√2
2
1
Other Inverse Trigonometric Functions
Function
Domain
Range
Graph
arccosine
arctangent
arcsine
Example 3: Find the exact value.
√3
a) arccos 2
b) cos-1(-0.5)
c) arctan(1)
d)tan-1 3
√3
Example 4: Use a calculator to approximate the value, if possible.
a) arctan 4.84
b) arcos (-.349)
c) sin-1(-1.1)
Example5: If possible, find the exact value.
a) tan(arctan(-14)
b) sin(arcsin π)
Example 6: Find the exact value.
a) cos[arctan(-3/4)]
b) sin[arccos(2/3)]
Example 7: Write each of the following as an algebraic expression for x.
a) sec(arctan x)
b) tan(arccos 2x)
c) cos[arccos(0.54)]
4.8 Applications and Models
What are we learning?
-Solving real life problems involving right triangles.
-Solving real life problems involving directional
bearings.
Why are we learning it?
-Trigonometric functions can be used in many real
world situations.
Solving a Right triangle
Example 1: Solve for the unknown measurements given that a = 5.46 and B = 70⁰.
A
c
b
B
a
C
Example 2: A ladder 16 feet long leans against the side of a house. Find the height h from the top of the ladder to the
ground if the angle of elevation of the ladder is 74⁰.
Example 3: From a point 65 feet in front of a church, the angles of elevation to the base of the steeple and the top of the
steeple are 35⁰ and 43⁰, respectively. Find the height of the steeple.
Example 4: From the time a small airplane is 100 feet high and 1600 ground feet from its landing runway, the plane
descends in a straight line to the runway. Determine the plane’s angle of descent.
Trigonometry and Bearings
In surveying and navigation, directions are generally given in terms of bearings. A bearing measures the acute angle that
a path or line of sight makes with a fixed north-south line.
Example 5: A sailboat leaves a pier and heads due west at 8 knots. After 15 minutes the sailboat tacks, changing course
to N 16⁰ W at 10 knots. Find the sailboat’s bearing and distance from the pier after 12 minutes on this course.