Download Exploration 4-2a: Properties of Trigonometric Functions

Name:
Group Members:
Exploration 4-2a: Properties of
Trigonometric Functions
Date:
Objective: Use the properties of trigonometric functions to transform an expression to
another given form.
1. Write the four trigonometric functions tan x, cot x,
sec x, and csc x in terms of sin x and cos x.
6. Divide both sides of the equation in Problem 5 by
cos2 x. Simplify the result to get a Pythagorean
property relating sec x and tan x.
2. The property sec x = cos1 x is called a reciprocal
property. Why do you think this is the property’s
name?
sin x
3. What is the name of the property tan x = cos
x?
7. Derive a Pythagorean property relating csc x and
cot x.
4. Evaluate cos2 0.6 C sin2 0.6. Sketch a 0.6-radian angle
in standard position, and use the drawing to explain
the significance of your answer.
8. Derive another quotient property expressing tan x in
terms of sec x and csc x.
5. What is the name of the property cos2 x + sin2 x = 1?
(Over)
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Exploration 4-2a: Properties of
Trigonometric Functions continued
9. Transform the expression csc x tan x to sec x. Start by
writing the given expression. Then substitute using
appropriate properties and simplify.
10. The expression csc x tan x in Problem 9 involves two
functions. The answer, sec x, involves only one
function. What could be your thought process in
deciding how to start this problem?
68 / Exploration Masters
Date:
11. Transform csc x tan x cos x to 1.
12. What did you learn as a result of doing this
Exploration that you did not know before?
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Exploration 4-3a: Transforming an Expression
Date:
Objective: Use the trigonometric properties to transform an expression to another
given form.
1. Write the three reciprocal properties.
3. Write the three Pythagorean properties.
2. Write the two quotient properties.
4. Transform sec x D cos x to sin x tan x. Beside each step, write the technique you used from
your “list of things to try.”
5. What did you learn as a result of doing this Exploration that you did not know before?
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Exploration 4-3b: Trigonometric Transformations
Date:
Objective: Use the properties of functions of one argument to transform trigonometric
expressions.
1. Without consulting text, notes, or other students,
write
4. Transform csc P cos2 P C sin P to csc P.
(Try factoring out the csc P you want, first.)
• The three reciprocal properties:
• The two quotient properties:
• The three Pythagorean properties:
5. On your grapher, plot y1 = tan2 x and y2 = sec2 x. Use
radian mode and a window with an x-range of [0, π]
and a y-range of [0, 10]. By appropriate tracing, find
out how the two graphs are related to each other.
How does this relationship correspond to the
trigonometric properties? Check your graph with
your instructor.
2. Transform (1 C cos A)(1 D cos A) to sin2 A.
6. What did you learn as a result of doing this
Exploration that you did not know before?
3. Transform cot B C tan B to csc B sec B.
70 / Exploration Masters
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Exploration 4-3c: Trigonometric Identities
Date:
Objective: Use the trigonometric properties to transform an expression to another given
form.
No books or notes!
csc x
5. Quick: Why does cot x = sec
x?
1. Write the three reciprocal properties.
2. Write the two quotient properties.
6. Prove that
sec A
sin A
sin A
D cos
A = cot A is an identity.
3. Write the three Pythagorean properties.
7. Prove:
4. Prove the alternate form of the quotient property,
tan x =
sec x
csc x
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1
1 D cos B
+ 1 + 1cos B = 2 csc2 B
8. What did you learn as a result of doing this
Exploration that you did not know before?
Exploration Masters / 71
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Exploration 4-4a: Arccosine, Arcsine, and Arctangent
Date:
Objective: Find values of arccosine, arcsine, and arctangent by calculator.
1. Find the degree measure of arccos 0.4 that equals
cosD1 0.4. Sketch the angle in a uv-coordinate system
(draw the arc and the arrow). Show the reference
triangle and label two of its sides.
6. Write the general solution for arcsin 0.3.
v
u
7. Find sinD1 (D0.8) and a value of arcsin (D0.8)
terminating in a different quadrant. Sketch the
angles and label each reference triangle.
v
u
2. On the same axes, sketch another angle arccos 0.4 in
a different quadrant. Sketch and label the reference
triangle. Find the degree measure of the angle.
8. Find tanD1 5 and a value of arctan 5 terminating in a
different quadrant. Sketch the angles and label each
reference triangle.
v
3. Write the general solution for arccos 0.4.
u
9. Write the general solution for arctan 5.
4. Find the value of arcsin 0.3 that equals sinD1 0.3.
Sketch the angle. Label two appropriate sides of the
reference triangle.
v
10. Find two values of arctan (D0.6) terminating in two
different quadrants. Sketch the angles and label each
reference triangle.
u
v
u
5. Sketch another angle arcsin 0.3 in a different
quadrant. Sketch and label the reference triangle.
Find the measure of the angle.
11. What did you learn as a result of doing this
Exploration that you did not know before?
72 / Exploration Masters
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Exploration 4-4b: Trigonometric Equations
Date:
Objective: Solve equations in which trigonometric functions appear.
1. Solve: 2 cos (θ D 17−) = 1, θ E [0−, 720−]
4. The figure shows the graphs of
y1 = D1 D 5 sin θ y2 = 2 cos2 θ
2. Solve: tan2 θ D 2 tan θ D 3 = 0, θ E [D360−, 360−]
Show on the graph that all of your answers in
Problem 3 are correct.
y
5
180°
90°
270°
450°
630°
θ
720°
5
5. Explain how what you have been studying about
transformation of trigonometric expressions allows
you to solve trigonometric equations.
3. Solve: D1 D 5 sin θ = 2 cos2 θ, θ E [D180−, 720−]
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Exploration 4-5a: Parametric Function
Pendulum Problem
Date:
Objective: Given a pendulum moving in both the x- and y-directions, predict its position,
(x, y), at any given time, t.
A pendulum hangs from the ceiling. The point on the
floor above which the pendulum bob is situated when it
is at rest is the origin, (0, 0), of a Cartesian coordinate
system. The x- and y-axes run parallel to the walls of the
room and the x-y plane is horizontal.
5. Using your equations in Problem 4, make a table of
values of x and y for various values of t from 0
through 1 complete orbit of the pendulum.
t
x
y
1. Give the pendulum bob a displacement of 30 cm
in the positive x-direction. Let it go and time its
period. Sketch the graph of x as a function of time,
t seconds, since it was released.
6. Plot the points (x, y) and connect them.
2. Starting with the pendulum bob at (0, 0), give it a
push in the y-direction just hard enough to make
it swing with an amplitude of 20 cm. Does the
pendulum have the same period this time? Sketch
the graph of y as a function of t.
y
25
x
40
3. Starting with the pendulum bob at (30, 0), give it a
push in the y-direction just hard enough to make it
have an amplitude of 20 cm in the y-direction, as in
Problem 2. Sketch. Describe the path followed by the
pendulum bob.
7. What geometrical figure describes the graph in
Problem 6?
8. What special name is given to a variable, such as t
in this problem, upon which two or more other
variables depend?
9. What special name is given to functions in which two
or more variables depend on the same independent
variable?
4. Assume that the graphs in Problems 1 and 2 are
sinusoids. Write particular equations for x and y as
functions of t.
xH
10. Put your grapher in parametric mode. Then enter the
equations for x and y from Problem 4. Pick suitable
ranges for x, y, and t. Have the calculator plot the
graph. (Use ZOOM SQUARE to get equal scales on the
axes.) Does the graph look like the graph you plotted
in Problem 6?
11. Just for fun, see if you can transform the two
equations from Problem 4 into one equation in only
x and y by eliminating the parameter t.
yH
74 / Exploration Masters
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Exploration 4-5b: Parametric Equations for Ellipses
Date:
Objective: Given a figure involving ellipses, write parametric equations and draw the
graph on your grapher.
4. Frustum of a cone (“truncated” cone):
y
6
y
6
x
10
x
10
1. The figure shows an ellipse with center at (5, 3),
x-radius H 4, and y-radius H 2. Write what you think
the parametric equations of this ellipse are.
xH
Top:
x1 H
y1 H
yH
2. Plot these parametric equations on your grapher. Use
a window with equal scales on both axes. Does the
graph agree with the figure?
For Problems 3–5, write parametric equations for each
ellipse shown. Plot the ellipses using degree mode with
t E [0−, 360] and a t-step of 10−. Use appropriate Boolean
variables and dot style for the parts of the ellipses that
are hidden. Then use the DRAW command to draw line
segments and dots in appropriate places.
Bottom, solid (use a Boolean variable)
x2 H
y2 H
Bottom, dashed (use a Boolean variable)
x3 H
y3 H
5. Cone oriented the other way:
y
3. Cone:
6
y
6
x
10
x
10
Solid (use a Boolean variable)
xH
x1 H
yH
y1 H
Dashed (use a Boolean variable)
x2 H
y2 H
6. What did you learn as a result of doing this
Exploration that you did not know before?
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Exploration 4-6a: Graphs of Inverse
Trigonometric Relations
Date:
Objective: Investigate the graphs of inverse circular functions and relations by plotting
them on your grapher.
4. Come back to function mode. Plot the graph of the
function y H sinD1 x. Because the calculator gives
only the principal value of the inverse sine, the
graph is called the principal branch. Darken the part
of the given graph that corresponds to the principal
branch.
This graph shows the inverse circular relation
y = arcsin x
In this Exploration, you will learn how to plot this graph
on your grapher.
y
5. Return to the parametric mode and enter the parent
sine function graph. Enter
x2T = t
y2T = sin t
1
x
1
Keep the x1T, y1T graph active. Sketch the resulting
graph on the given figure.
6. What relationship do the sine and inverse sine
graphs have to each other? How do they relate to the
line y H x?
1. Recall that y H arcsin x means that x H sin y.
Calculate x for y H 4.
2. Write the general solution for y H arcsin x. Use it to
find the four values of arcsin 0.4 that are on the
portion of the graph shown. Round to one decimal
place. Plot points on the graph corresponding to the
four answers.
7. Plot the graphs of y H cos x and y H arccos x on your
grapher. Use the same window as in the previous
problems. Sketch the result here, along with the
line y H x.
y
1
x
1
3. Reproduce the given graph on your grapher. A clever
way to do this is to use parametric mode. Enter the
equation as
x1T = sin t
y1T = t
Use a t-range of D9 to 9. Use [D9, 9] for the
x-range and equal scales for the two axes. Check the
graph with your instructor.
76 / Exploration Masters
8. What did you learn as a result of doing this
Exploration that you did not know before?
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Exploration 4-6b: Principal Branches of Inverse
Trigonometric Relations
Date:
Objective: Figure out the principal branches of each of the six inverse circular functions.
1. On your grapher, plot the inverse circular function
y H sinD1 x. Use equal scales on the two axes. On the
graph of y H arcsin x shown here, darken the
principal branch of the inverse sine relation on the
part of the graph that is the inverse sine function.
Give the range of the principal branch.
4. Use the technique in Problems 1 and 2 to find the
range of y H tanD1 x. Darken the principal branch on
this graph of y H arctan x.
Range:
y
Range:
y
1
x
1
1
x
1
2. Use the technique in Problem 1 to find the range of
y H cosD1 x. Darken the principal branch on the graph
of y H arccos x, shown next.
5. The range of y H sinD1 x is the closed interval CD π2 , π2 D.
Explain why the range of y H tanD1 x cannot include
the endpoints.
Range:
y
1
x
1
3. Why can’t the range of the inverse cosine function
(y H cosD1 x) be the same as the range of the inverse
sine function (y H sinD1 x)?
6. If the range of the inverse tangent function
(y H tanD1 x) were [0, π] (excluding π2 ), like the range
of y H cosD1 x, then y H tanD1 x would still be a
function. What disadvantage would there be to
defining the range of y H tanD1 x this way?
(Over)
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Exploration 4-6b: Principal Branches of Inverse
Trigonometric Relations continued
Date:
7. Duplicate the previous graph of y H arctan x on your
grapher. Give the parametric equations you used.
Check your graph with your instructor.
10. Look in the text to find out if the principal branch
you chose in Problem 9 is the commonly accepted
one.
8. The graph here shows y H arccot x. How can you
define the range of the function y H cotD1 x in such a
way that the function is continuous? Darken this
principal branch of y H arccot x.
11. This next graph shows y H arccsc x. Shade what you
think the principal branch is. Write the range of the
function you shaded.
Range:
Range:
y
y
1
1
x
x
1
1
12. Does your answer to Problem 11 agree with the range
listed in the text?
9. This next graph shows y H arcsec x. There is no way
to restrict the range to make a continuous function
y H secD1 x and still use all of the domain. Darken
what you think would be the best choice for the
principal branch. Write the range.
Range:
y
13. Look up in the text the five criteria for picking the
ranges of the principal branches of the six inverse
circular functions. Write the criteria here.
1
x
1
14. What did you learn as a result of doing this
Exploration that you did not know before?
78 / Exploration Masters
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Exploration 4-6c: Inverse Trigonometric
Relation Values
Date:
Objective: Calculate multiple values of inverse circular relations and confirm them
graphically.
y
y
y
10
10
5
5
10
5
x
5
x
2
2
5
x
2
2
5
5
5
1. Evaluate arcsin 0.8 for all values that show on the
graph of y H arcsin x at the top of this column.
Round final answers to the nearest 0.1 unit. Mark
the solutions on the graph.
2. Evaluate arccos (D0.3) for all values that show on the
graph of y H arccos x at column top. Round final
answers to the nearest 0.1 unit. Mark the solutions
on the graph.
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3. Evaluate arctan 4 for all values that show on the
graph at the top of this column, of y H arctan x.
Round final answers to the nearest 0.1 unit. Mark
the solutions on the graph.
4. Darken the principal branch of each graph. Circle
the point your calculator gives for sinD1 0.8,
cosD1 (D0.3), and tanD1 4 on the respective graphs,
showing in each case that the value is on the
principal branch.
5. What did you learn as a result of doing this
Exploration that you did not know before?
Exploration Masters / 79
4. 3 radians = 3 •
5. 240− = 240− •
4. cos2 0.6 + sin2 0.6 = 1
180−
= 171.8873…
π
v
4π
π
=
radians
180−
3
sin 0.6
1
7
6.
radians
10
u
cos 0.6
7.
y
5
5. Pythagorean property
3
6. cos2 x + sin2 x = 1
1
1
• (cos2 x + sin2 x) = 2
cos2 x
cos x
cos2 x sin2 x
1
+
=
cos2 x cos2 x cos2 x
sin x 2
1 2
=
1+
cos x
cos x
1 + tan2 x = sec2 x
x
5
8. y = 5 D 2 cos
10
15
20
25
π
π
x or y = 5 + 2 cos (x D 4)
4
4
9. y(27) = 6.4142… m M 6.4 m
10. x = 14.9872… m M 15.0 m
11. 7 feet
π
(x D 1) = 7 ⇒ x = 1 a.m.
5.8
Period = 2 • 5.8 = 11.6 hours
3 + 4 cos
π
(x D 1) = D1 ⇒ x = 6.8 = 6:48 a.m.
5.8
1 foot deep
12. 3 + 4 cos
13. 4:00 p.m. is x = 16; y(16) = 1.9298… ft M 1.9 ft, which agrees
with the graph.
14. Because this happens at the end of the second complete
π
cycle, it is where 5.8
(x D 1) = 4π ⇒ x = 24.2 hr = 12:12 a.m.
on January 2.
15. 5.465… hr ≤ x ≤ 8.1343… hr or approximately
5:28 a.m. ≤ x ≤ 8:08 a.m.
5.8
D3
π
(x D 1) = 0 ⇒ x = 1 +
cosD1
5.8
π
4
= 5.4656… hr M 5:28 a.m.
7. cos2 x + sin2 x = 1
1
1
• (cos2 x + sin2 x) = 2
sin2 x
sin x
cos2 x sin2 x
1
+
=
sin2 x sin2 x sin2 x
cos x 2
1 2
b
+1=
sin x
sin x
cot2 x + 1 = csc2 x
8. tan x =
sin x 1/cos x sec x
=
=
cos x 1/sin x csc x
9. csc x • tan x =
1
sin x
1
•
=
= sec x
sin x cos x cos x
10. Answers will vary.
1
sin x
•
• cos x
sin x cos x
1
=
• cos x = 1
cos x
11. csc x • tan x • cos x =
16. 3 + 4 cos
17. Answers will vary.
Chapter 4 • Trigonometric Function
Properties, Identities, and Parametric
Functions
Exploration 4-2a
sin x
cos x
cos x
cot x =
sin x
1
sec x =
cos x
1
csc x =
sin x
1. tan x =
12. Answers will vary.
Exploration 4-3a
1
tan x
1
sec x =
cos x
1
csc x =
sin x
1. cot x =
sin x sec x
=
cos x csc x
cos x csc x
cot x =
=
sin x sec x
2. tan x =
3. cos2 x + sin2 x = 1
1 + tan2 x = sec2 x
cot2 x + 1 = csc2 x
2. One function is the reciprocal of the other.
3. Quotient property
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Solutions to the Explorations / 243
4. sec x D cos x ⇒ sin x tan x
sec x D cos x
1
=
D cos x
cos x
Exploration 4-3c
1. sec x =
1
1
1
, csc x =
, cot x =
cos x
sin x
tan x
Make fractions to add for 1 term.
cos2 x
1
D
=
cos x cos x
2. tan x =
sin x
cos x
, cot x =
cos x
sin x
Get common denominator.
1 D cos2 x
=
cos x
4. tan x =
3. cos2 x + sin2 x = 1, tan2 x + 1 = sec2 x, cot2 x + 1 = csc2 x
sin x
cos x
1/csc x
=
1/sec x
sec x
=
csc x
Add to get 1 term.
sin2 x
=
cos x
When you see squares of functions, think Pythagorean.
sin x
= sin x •
cos x
If you see something you want in the answer, guard it with
your life.
sin x • tan x
Familiar property
5. cot x =
6.
5. Answers will vary.
Exploration 4-3b
1
1
1
, csc x =
, cot x =
cos x
sin x
tan x
sin x
cos x
tan x =
, cot x =
cos x
sin x
cos2 x + sin2 x = 1, tan2 x + 1 = sec2 x,
cot2 x + 1 = csc2 x
1. sec x =
2. (1 + cos A)(1 D cos A) = 1 D cos2 A = sin2 A
cos B sin B
+
sin B cos B
2
cos B
sin2 B
=
+
sin B cos B sin B cos B
cos2 B + sin2 B
=
sin B cos B
1
=
sin B cos B
= csc B sec B
3. cot B + tan B =
1
sin P )
csc P
2
2
= csc P (cos P + sin P )
= csc P
4. csc P cos2 P + sin P = csc P (cos2 p +
7.
1
1
csc x
=
=
tan x sec x/csc x sec x
sin2 A
sec A sin A 1/cos A
D
=
D
sin A cos A
sin A
cos A sin A
1
sin2 A
=
D
cos A sin A cos A sin A
1 D sin2 A
=
cos A sin A
cos2 A
=
cos A sin A
cos A
=
sin A
= cot A
1
1
+
1 D cos B 1 + cos B
1 + cos B
1 D cos B
=
+
(1 + cos B)(1 D cos B) (1 D cos B)(1 + cos B)
1 + cos B
1 D cos B
=
+
1 D cos2 B 1 D cos2 B
1 + cos B + 1 D cos B
=
1 D cos2 B
2
=
sin2 B
= 2 csc2 B
8. Answers will vary.
Exploration 4-4a
1. arccos 0.4 H 66.4218…−
v
1
5.
θ
y
u
0.4
5
y2
y1
x
π
2
y2 appears to be 1 + y1. This is consistent with the property
sec2 x = tan2 x + 1.
6. Answers will vary.
244 / Solutions to the Explorations
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©2003 Key Curriculum Press
2. arccos 0.4 H Y66.4218…−
v
9. arctan 5 = 78.6900…− + n180−
or 158.6900…− + n360−
10. arctan (D0.6) = D30.9637…− or 149.6362…−
1
v
θ
u
θ
1
0.4
1
u
θ
θ
1
3. arccos 0.4 H J66.4218…− C 360n−
Original and new θ have the same reference angle.
4. arcsin 0.3 H 17.4576…−
11. Answers will vary.
v
0.3
1
θ
1. 2 cos (θ D 17−) = 1 ⇒ cos (θ D 17−) =
u
5. arcsin 0.3 H 162.5423…−
v
0.3
0.3
1
θ
1
θ
Exploration 4-4b
u
6. arcsin 0.3 H 17.4576…− C n360−
or 162.5423…− C n360−
1
2
⇒ θ D 17− = ±60 + n360−
⇒ θ = 53− + n360− or D77− + n360−
= 53−, 283−, 413−, 643−
2. tan2 θ D 2 tan θ D 3
= (tan θ D 3)(tan θ + 1) = 0
⇒ tan θ = 3 or tan θ = D 1
⇒ θ = 71.5650…− + n180− or
θ = D 45− + n180−
θ = D288.4349…−, D225−, D108.4349…−,
D45−, 71.5650…−, 135−, 251.5650…−, 315−
3. D1 D 5 sin θ = 2 cos2 θ = 2 D 2 sin2 θ
⇒ 2 sin2 θ D 5 sin θ D 3 = 0
⇒ (2 sin θ + 1)(sin θ D 3) = 0
⇒ sin θ = D 12 (Note: sin θ < 3 for all θ)
⇒ θ = D30− + n360− or
θ = D150− + n360−
θ = D150−, D30−, 210−, 330−, 570−, 690−
4.
7. sinD1 (D0.8) H D53.1301…−
arcsin (D0.8) H 180− D (D53.1301…−)
H 233.1301…−
y
5
v
θ
90°
0.6
0.8
0.6
θ
θ
1
1
u
270°
180°
450°
630°
720°
5
0.8
5. Answers will vary.
8. tan D15 = 78.6900…−
arctan 5 = 78.69000…− + 180−
= 158.6900…−
Exploration 4-5a
1. Example
v
x (cm)
1
30
θ
u
t (sec)
p
θ
1
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30
Solutions to the Explorations / 245
10.
2. Example
y
y (cm)
30
10
t (sec)
x
10
p
30
Period should be the same.
2π
x
2π
y
t = , sin
t=
P
30
P
20
2π
2π
cos2
t + sin2
t=1
P
P
2 2
x
y
⇒
+
=1
30
20
11. cos
3. Path is an ellipse.
4. If the period is P seconds, then
2π
x = 30 cos
t
P
2π
t
y = 20 sin
P
Exploration 4-5b
5.
t
x
0
30
P
8
21.2132…
P
4
0
0
3P
8
D21.2132…
P
2
D30
5P
8
D21.2132…
3P
4
1. x H 5 C 4 cos t
y H 3 C 2 sin t
y
2. Equations are correct.
14.1421…
3. x H 4 C 2 cos t, y H 5 C 0.4 sin t
20
4. Top:
x H 4 C 2 cos t, y H 5 C 0.4 sin t
14.1421…
0
D14.1421…
D20
0
7P
8
21.2132…
P
30
D14.1421…
0
6.
Bottom: (solid)
x H 4 C 3 cos t/(t L 0 and t K π),
y H 1 C 0.6 sin t/(t L 0 and t K π)
Bottom: (dashed)
x H 4 C 3 cos t/(t L π and t K 2π),
y H 1 C 0.6 sin t/(t L π and t K 2π)
5. (solid)
x H 1 C 0.4 cos t/(t L π2 and t K 3π
2 ),
y H 3 C 2 sin t/(t L π2 and t K 3π
2 )
(dashed)
x H 1 C 0.4 cos t/(t L D π2 and t K π2 ),
y H 3 C 2 sin t/(t L D π2 and t K π2 )
6. Answers will vary.
y
2
Exploration 4-6a
x
40
1. x H sin 4 H D0.7568…
2. arcsin x H sinD1 x C 2πn or (π D sinD1 x) C 2πn
y H arcsin 0.4 H 0.4115… C 2πn or 2.7300… C 2πn
y H D5.8716…, D3.5531…, 0.4115…, 2.7300…
y ≈ D5.9, D3.6, 0.4, 2.7
y
7. The graph is an ellipse.
8. Parameter
9. Parametric function
246 / Solutions to the Explorations
1
x
1
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1
©2003 Key Curriculum Press
4. Range: D π2 ≤ y ≤ π2
3.
y
y
1
x
1
1
x
1
4.
y
1
x
5. tan π2 and tan D π2 are undefined. So the range of y H tanD1 x
cannot include these numbers.
( )
1
6. The graph would not be continuous.
7. x H tan 5; y H t
5.
y
8. Range: 0 < y < π
y
1
x
1
1
6. They are inverses of each other. Their graphs are reflections
across the line y H x.
x
1
7.
y
1
9. Range: 0 ≤ y ≤ π, and y ≠ π2
x
y
1
1
8. Answers will vary.
x
1
Exploration 4-6b
1. Range: D π2 ≤ y ≤ π2
y
10. It is the commonly accepted branch.
1
11. Range: D π2 ≤ y ≤ π2 , and y ≠ 0
x
y
1
2. Range: 0 ≤ y ≤ π
1
y
x
1
1
x
1
12. Agrees
D1
D1
3. If the range of cos were the same as that of sin , then
cosD1 would not be a function.
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1
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Solutions to the Explorations / 247
13. 1. Must be a function
2. Must use entire domain
3. arctan 4 H 1.3258… C πn
≈ D5.0, D1.8, 1.3, 4.5, 7.6, 10.7
y
3. Should be continuous
4. Should be centrally located
10
5. If there is a choice, choose the positive branch.
14. Answers will vary.
Exploration 4-6c
5
1. arcsin 0.8 H 0.9272… C 2πn or 2.2142… C 2πn
≈ D5.4, D4.1, 0.9, 2.2, 7.2, 8.5
y
x
5
5
10
5
5
4. The values are on the principal branches.
5. Answers will vary.
x
2
2
Chapter 5 • Properties of
Combined Sinusoids
Exploration 5-2a
1. y1 is the solid graph, y2 the dashed graph.
5
2.
y
2. arccos (D0.3) H ±1.8754… C 2πn
≈ D44, D1.9, 1.9, 4.4, 8.2, 10.7
5
θ
y
360°
10
3. A = 5
D = 53.1301…−
4. y4 = 5 cos (θ D 53.1301…−)
The graph coincides.
y
5
5
θ
360°
x
2
2
5
5. A cos D = 3
A sin D = 4
A = √A2 = √A2(cos2 D + sin2 D)
= √(A cos D)2 + (A sin D)2
= √32 + 42
=5
6. 3 = A cos D = 5 cos D ⇒ cos D =
4 = A sin D = 5 sin D ⇒ sin D =
D = cosD1 35 = sinD1 45 = 53.1301…−
248 / Solutions to the Explorations
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1
©2003 Key Curriculum Press