Download Chapter 6 Extending Periodic Functions

Chapter 6
Extending Periodic Functions
Chapter Outline
Section 6.1 You will solve trig equations by looking at a graph, the unit circle, and using the
inverse functions on your calculator. You will see how several trig equations and applications
have more than one solution. You will also investigate the inverse trig functions.
Section 6.2 You will be extending your modeling of periodic functions to more complex
situations. You will see how a trig function with two different angles can be simplified.
Section 6.3 You will work with other trig formulas and identities to solve more complex trig
equations.
Section 6.4 You will see how the motion of a spring can be modeled by using a
combination of a sinusoidal and an exponential function.
Math Notes: Solving Periodic Functions
When solving periodic functions (such as 𝑠𝑖𝑛 𝑥 = −
), we get an infinite number of
solutions (or no solution). In the example 𝑠𝑖𝑛 𝑥 = −
, we get the solutions:
𝑥= …−
,
, … and 𝑥 = … −
,
+2𝜋𝑛,
This may be written more compactly as x =
,
,
,…
+2𝜋𝑛, where 𝑛 is any integer.
If we restrict the domain to [0, 2𝜋), we get only two solutions: 𝑥 =
and
.
Math Notes: Notation for Inverse Trigonometry
There are two standard notations to represent the inverse for trig functions. The first uses
inverse notation. For example:
𝑦 = 𝑠𝑖𝑛 −1 𝑥 𝑎𝑛𝑑 𝑦 = 𝑐𝑜𝑠 −1 𝑥
You must be careful with this notation. The function 𝑦 = 𝑠𝑖𝑛 −1 𝑥 is not the same
function as
𝑦=
1
𝑠𝑖𝑛 𝑥
. The function,
sin x, while the function 𝑦 = 𝑠𝑖𝑛
−1
𝑦=
1
𝑠𝑖𝑛 𝑥
is 𝑦 = 𝑐𝑠𝑐 𝑥, the reciprocal function for
𝑥 is called the inverse function for 𝑠𝑖𝑛 𝑥.
Another form is using 𝑦 = 𝑎𝑟𝑐𝑠𝑖𝑛 𝑥 for the inverse of sine and 𝑦 = 𝑎𝑟𝑐𝑐𝑜𝑠 𝑥 for the
inverse of cosine. This notation, 𝑦 = 𝑎𝑟𝑐𝑠𝑖𝑛 𝑥 can be thought of as “the arc whose sine is
…” This connects the inverse back to the unit circle.
Both 𝑦 = 𝑠𝑖𝑛 −1 𝑥 and 𝑦 = 𝑎𝑟𝑐𝑠𝑖𝑛 𝑥 mean exactly the same thing.
𝑦 = 𝑐𝑜𝑠 −1 𝑥 and 𝑦 = 𝑎𝑟𝑐𝑐𝑜𝑠 𝑥 are two different ways to write the exact same thing.
Math Notes: Solving SSA
When solving a triangle with two sides and a non-included angle (SSA), you need to be
aware that the triangle can have no solution, one solution, or two solutions. If you use the
Law of Sines to solve this type of triangle, you will need to find the missing angle first.
Case 1
Here, side
a is too
short to
make a
triangle; no
matter how
we swing it
from B, it
won’t reach
the other
side.
Case 2
Here, side
a is too
long. When
we swing it
from B, it
hits the
opposite
side in two
places:
Cand C′.
ΔABC and
ΔABC′ both
have a 30°
angle and
sides of 10
and a.
Case 3
Here, side
a is just
right. When
we swing it
from B, it
touches the
opposite
side at just
one point,
called C.
There will
be just one
triangle,
ΔABC.
Math Notes: Angle of Inclination
The tangent of the angle of inclination is the slope of a
line. So the line y = mx + b makes an angle θ with the xaxis where tan θ = m.
Note that θ is always acute. If the slope of the line is
negative, the angle θ will be negative.
Math Notes: 5-Point Graphing Method Reminder
When graphing a sine or cosine function, it helps to locate 5 key
points that describe the graph.
These points are shown on the graph as black dots.
𝜋
For example: 𝑦
= 2𝑐𝑜𝑠( (𝑥 − 2)) + 1.
3
The 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 = 2. The graph is shifted up 1 and right 2.
Use the amplitude and the shifts to determine the first point.
𝜋
Angular frequency is , so the period is found by
3
𝜋
𝑝 ⋅ = 2𝜋, so 𝑝 = 6.
3
Divide the period into 4 equal pieces to determine the spacing between points.
Plot the remaining four points by moving to the middle, then bottom, back to the middle and
finally back to the top.
Math Notes: General Equation for Cosine
If 𝑦 = 𝑐𝑜𝑠 𝑥 is stretched vertically by a, has an angular frequency of b, is shifted h units to
the right, and k units up, the resulting graph has the equation:
𝑦 = 𝑎 𝑐𝑜𝑠(𝑏(𝑥 − ℎ)) + 𝑘.
The amplitude is a . The period (p) and angular frequency (b) are related by 𝑝
· 𝑏 = 2𝜋.
Math Notes: Sum and Difference Formulas
Although sine and cosine do not distribute over sums or differences, it turns out that sin(α ± β) and
cos(α ± β) can be expressed in terms of sin α, cos α, sin β, and cos β. In this and future math courses,
you will need to use these identities many times to rewrite sin(α ± β) and cos(α ± β).
Angle Sum Identities
sin(α + β) = sin α cos β + cos α sin β
cos(α + β) = cos α cos β − sin α sin β
Angle Difference Identities
sin(α − β) = sin α cos β − cos α sin β
cos(α − β) = cos α cos β + sin α sin β
Combined Forms
sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β ∓ sin α sin β
Math Notes: Double-Angle Formulas for Sine and Cosine
cos 2α = cos2 α − sin2 α
OR = 2cos2 α − 1
OR = 1 − 2 sin2 α
sin 2α = 2 sin α cos α
Math Notes: Half-Angle Formulas for Sine and Cosine
The half-angle formulas are listed below. You must determine which sign to use ( + or −) from the
𝜃
quadrant in which lies.
2
Also useful, are the formulas you found as preliminary steps to the half angle formula:
and
Chapter 6 Closure Problems with Answers Included for Self-Checking
CL 6-162. Explain why 𝑠𝑖𝑛 𝑥
1
= 2 has more than one solution.
a. Use the unit circle to justify your answer.
b. Use a graph to justify your answer.
c.
What does 𝑠𝑖𝑛
−1
1
1
(2) equal? How is this different from 𝑠𝑖𝑛 𝑥 = 2 ?
CL 6-163. Solve each of the following equations on the domain [0, 2𝜋].
a. 2 𝑐𝑜𝑠 𝑥 + 1 = 0
b. 𝑠𝑖𝑛
2
3
𝑥=4
c. 𝑡𝑎𝑛 𝑥 = −1
CL 6-164. Find a function in terms of sine and another in terms of cosine that will generate the graphs
below.
a.
b.
CL 6-165. Given ΔABC with ∠ A = 34°,
= 8cm, and
= 6cm.
a. Draw two different possible triangles that would have the measurements described for
ΔABC. Why are there two possible solutions?
b. Find two possible lengths for side
.
c. What is the difference in the area of the two triangles?
CL 6-166. Find the angle of inclination for each line below.
2
a. 𝑦 = 3 𝑥 − 1
b. 𝑦 = −2𝑥 + 5
c.
What is the measure of the acute angle that is formed at the intersection of the two lines?
CL 6-167. 𝑦 = 𝑠𝑖𝑛 −1 𝑥, 𝑦 = 𝑐𝑜𝑠 −1 𝑥, and 𝑦
have restricted ranges. Give the range of each of them.
CL 6-168. Given: 𝑠𝑖𝑛 𝐴
5
= 𝑡𝑎𝑛
5
−1
𝑥 are all functions because they
𝜋
= 13and 𝑠𝑒𝑐 𝐵 = 4, where 0 < 𝐴, 𝐵 < 2 .
a. What is the exact value of cos A?
b. What is the exact value of sinB?
c. Find the exact value of cos(A + B).
CL 6-169. Simplify each of the following trig expressions.
a. 20 𝑠𝑖𝑛(2𝑥) 𝑐𝑜𝑠(2𝑥)
b. 𝑠𝑖𝑛(𝜋 − 𝑥)
c. 𝑠𝑖𝑛 2 𝜃 − 𝑐𝑜𝑠 2 𝜃
𝜋
d. 𝑐𝑜𝑠 (𝑥 + 2 )
CL 6-170. Solve the following equations in the domain stated.
a. 2 𝑐𝑜𝑠 2 𝑥 + 𝑠𝑖𝑛 𝑥 = 2, [0, 2𝜋]
b. 2 𝑐𝑜𝑠 2 𝑥 + 𝑠𝑖𝑛 𝑥 = 2
c. 𝑠𝑖𝑛 𝑥 − 𝑠𝑖𝑛(2𝑥) = 0, [0, 2𝜋]
d. 𝑠𝑖𝑛 𝑥 − 𝑠𝑖𝑛(2𝑥) = 0
CL 6-171. A spring is hanging from the ceiling of a room. It is pulled and released so that the distance
from the floor with respect to time is a sinusoidal motion. At t = 2.5 seconds, the spring is at a minimum
from the floor, 1.5 feet. At t = 3.75 seconds, the spring is at a maximum, 6 feet. Find a trigonometric
equation that models the motion of the spring as a function of time.
a. Find the height of the spring from the floor at t = 3 seconds.
b. Find the first two times when the spring is 2 feet from the floor.
CL 6-172. Solve each of the following equations.
a.
b.
c.
d.
18(1.03) 𝑥 − 20 = 300
𝑙𝑜𝑔2 5 + 𝑙𝑜𝑔2 𝑥 = 3
8𝑥 2.7 = 160
𝑙𝑜𝑔3 (𝑥 + 5) − 𝑙𝑜𝑔3 (𝑥 − 1) = 1
CL 6-173. Find the equation of the line 𝑦 = 𝑎𝑥 + 𝑏 so that 𝑓(𝑥) is continuous.
CL 6-174. Check your answers using the following table. Which problems do you feel confident about?
Which problems were hard? Have you worked on problems like these in math classes you have taken
before? Use the table to make a list of topics with which you need help or more practice.
Answers and Support for Closure Problems
What Have I Learned?
Problem
Solution
Need Help?
CL 6-162.
Lesson 6.1.1 Math Note
Problems 6-1, 6-2, 6-9, 6-24,
and 6-25
a.
, the function sin−1 x can only have one output.
c.
CL 6-163.
CL 6-164.
b.
a.
,
b.
,
c.
a. y = 2sin(2x) + 2, y = 2cos(2(x −
−3sin(
(x −
)) −1, y = 3cos(
(x +
CL 6-165.
)) + 2,
Lessons 6.1.1 and 6.1.4
Problems 6-5, 6-10, 6-26,6-27,
6-48, 6-56, and 6-99
,
b. y =
Lesson 6.2.1 Math Note
Problems 6-65 to 6-69, 6-72,
6-79, and 6-85
)) − 1
Lesson 1.3.1
Lesson 6.1.3 Math Note
a.
b. 10.63 cm and 2.63 cm
c. 17.89 cm2
CL 6-166.
CL 6-167.
CL 6-168.
a. 33.7°
b. −63.4°
c. 82.9°
sin−1:
a.
b.
c.
Problems 6-38, 6-40, and6113
Lesson 6.1.4 Math Note
Problems 6-52, 6-53, and6-55
, cos−1: [0, π], tan−1:
Lessons 6.1.2 and 6.1.4
Problems 6-20, 6-22, 6-23, 629, 6-50, and 6-58
Lesson 4.1.2
6.2.2
Math Note Problems 6-81,
6-82, 6-83, 6-96, 6-98, 6-109,
and 6-138
CL 6-169.
a.
b.
c.
d.
10 sin 4x
sin x
−cos(2θ)
−sin x
CL 6-170.
a. 0, π, 2π,
b. 0, π,
c.
CL 6-171.
,
Lesson 6.3.2 Math Note
Problems 6-117, 6-119, and
6-129
,
,
0, π, 2π,
d. 0, π,
Lesson 6.2.2 Math Note
Lesson 6.3.1 Math Note
Problems 6-83, 6-87, 6-88,
6-95, 6-106, 6-111, and 6-114
, all + 2πn
,
, all + 2πn
y = 2.25 sin(0.8π(x − 3.125)) + 3.75
a. 3.055 feet
b. At 0.2704 and 2.2296 seconds
CL 6-172.
a.
b.
c.
d.
97.364
8/5
3.033
4
CL 6-173.
y = −x − 6
Lesson 6.3.2
Problems 6-11, 6-71, 6-84,
6-85, 6-92, and 6-93
Lesson 3.3.2 Math Note
Problems 3-122 to 3-126, and
6-103
Lesson 5.2.4 Math Note
Problem 5-93 and 5-100