UNIT ONE
ADVANCED TRIGONOMETRY
MATH 611B
15 HOURS
Revised Feb 26, 03
18
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Relating Graphs and Solutions (5.1)
Invite student groups to do the Getting Started and Warm Up exercises
on p.242-3 as selected in the Suggested Resources column.
B10 analyse and apply the
graphs of the sine and
cosine functions
Challenge student groups to do the Explore and Inquire on p.244.
Student groups should read and discuss p.245-247
C39 analyse tables and
graphs of sine and
cosine equations to
find patterns
Note To Teachers: This section is essentially systems of equations
composed of a sinusoidal and linear equation. The students might
develop a better understanding of the patterns occurring if they do the
“Relating Graphs and Solutions Worksheet (5.1)” at the end of the
unit.
It may be easier for students to have the graphing calculator in the
degree mode to do the worksheet.
19
Worthwhile Tasks for Instruction and Assessment
Relating Graphs and Solutions (5.1)
Relating Graphs and Solutions
(5.1)
Technology
Graph using a graphing calculator. Inductively find a formula
for the exact solutions:
a) sin 2 x =
1
2
Suggested Resources
Relating Graphs and Solutions
Worksheet (5.1)
b) sin 2 x = 1
Math Power 12 p.247 # 5,7,9
Applications
p.248 # 25, 28
Solution for (b):
20
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Solving Trig Equations (5.2)
Student groups should do the Explore and Inquire on p.249.
B11 derive, analyse and
apply angle and arclength relationships
using the unit circle
Student groups should review the concepts of:
< angles in standard position
< reference angles
< ratios for special angles 0°, 30°, 45°, 60° and 90°
C42 create and solve
trigonometric
equations
Invite student groups to read and discuss the examples on p.249-252.
Students will be expected to solve these problems both algebraically and
graphically.
Students should be exposed to two graphical methods for the following
example:
C48 solve trigonometric
equations with and
without graphing
technology
Find the solutions for 2cosx !1 = 0 for !B
# x # B.
Method 1:
2nd Calc 5:intersect
Method 2:
2nd Calc: 2:zero
So students should see that they can find the solutions by looking for the
intersection of two graphs or by looking for the zeros of the single
graph.
21
Worthwhile Tasks for Instruction and Assessment
Suggested Resources
Solving Trig Equations (5.2)
Solving Trig Equations (5.2)
Pencil/Paper
Solve 2sin2 + 1 = .5 algebraically and graphically for 2, 0 #
2 # B.
Math Power 12 p.252 #1-31 odd, 10,
18,20
Group Activity
Solve for x where 0 # x # 2B . Then give a general solution
for: 3sin2 x + 2sin x = 1.
22
Applications p.253#32(a),33(b),35(b)
Trig Equation Worksheet (5.2)
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Using Technology (5.3)
Student groups should do the Explore and Inquire on p.254.
B10 analyse and apply the
graphs of the sine and
cosine functions
Student groups should read and discuss the examples on p.255-256.
C48 solve trigonometric
equations with and
without graphing
technology
a) systems of equations:
The text does not make clear the fact that there are two graphical
methods that can be used. Looking at the example on p.254 we can use:
graph y = cos x, y = x and find the intersection point.
b) combine both functions into a single function y = cos x ! x and find
the zeros of the function.
Note to Teachers: This function could have been written as y = x ! cos x
yielding the same zero or solution.
23
Worthwhile Tasks for Instruction and Assessment
Suggested Resources
Using Technology (5.3)
Using Technology (5.3)
Technology
A ferris wheel has a diameter of 50m and turns at a rate of 1.5
revolutions per minute. The height of a seat above the ground
after “t” minutes can be described using h = 21 ! 25 cos 3Bt
. How long after the ride starts will your seat be 31 m off the
ground for the first time.
Math Power 12 p.256 #1-5, 8
Technology
Solve for 2, 0# 2 # 2B, the equation 2 sin θ + 1 = 0.5
Technology
The shown below shows y = cos2 and y = .5x ! 1 using the
window [!2B,2B,B/4] and [!2,2,1].
a) To what single equation does the graph provide the
solution?
b) Use the graph to give an approximate solution.
24
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Trigonometric Identities (5.4)
Student groups should do the Explore & Inquire on p.258. They should
then read and discuss p.259-263.
B12 derive and apply the
Reciprocal and
Pythagorean Identities
Students should appreciate the difference between an equation and an
identity. An equation is a statement that is true for a limited number of
values. An identity is a statement that is true for any value of the
variable.
This section uses the basic trig identities below to solve various
problems:
Quotient Identities
Pythagorean Identities
tan θ =
sin θ
cos θ
cot θ =
cos θ
sin θ
sin 2 θ + cos2 θ = 1
tan 2 θ + 1 = sec 2 θ
1 + cot 2 θ = csc 2 θ
Reciprocal Identities
sin θ ⋅ csc θ = 1
cos θ ⋅ sec θ = 1
tan θ ⋅ cot θ = 1
Suggestions for verifying Trig Identities:
< work with the more complicated side of the equation
< substitute one or more of the basic identities to simplify, factor or
multiply to simplify
< multiply expressions equivalent to one
< express trig functions in terms of sine and cosine
The properties of equalities do not apply, so that operations cannot be
performed on both sides of an unverified identity.
25
Worthwhile Tasks for Instruction and Assessment
Suggested Resources
Trigonometric Identities (5.4)
Trigonometric Identities (5.4)
Group Activity
Determine whether or not the following is a trigonometric
Math Power 12 p.264 #1-31 odd
omit # 7
identity:
7 sin θ + 5 cos θ
= 7 sec θ + 5 csc θ
sin θ cos θ
.
Performance
Demonstrate to your group/class the following. When an
object is fired with an initial velocity v0 at an angle of
elevation 2, its height y above the ground and its horizontal
displacement x are related by the equation:
y =
− gx 2
x sin θ
+
cos θ
2v0 2 cos2 θ
Rewrite this equation so that tan2 is the only trig function
appearing.
26
Trig Identity Worksheet (5.4)
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Sum, Difference, Double Angle Identities (5.5)
Challenge students to do the Investigation on p.266-7.
B13 explore and verify
other trigonometric
identities and solve
trigonometric
equations
B41 derive and apply the
compound angle
identities and the half
and double angle
identities
Students will not be expected to memorize the following identities.
Sum and Difference Identities
sin(A + B) = sinA cosB + cosA sinB
sin(A ! B) = sinA cosB ! cosA sinB
cos(A + B) = cosA cosB ! sinA sinB
cos(A ! B) = cosA cosB + sinA sinB
tan(A + B) = tanA + tanB
1 ! tanA tanB
tan(A ! B) = tanA ! tanB
1 + tanA tanB
Double Angle Identities
sin2A = 2sinA cosA
cos2A = cos2 A ! sin2 A
or 2cos2 A ! 1 or 1 ! 2sin2 A
tan2A =
where A … ± B/4 & B/2 + nB
2 tanA
1 ! tan2 A
27
Worthwhile Tasks for Instruction and Assessment
Sum, Difference, Double Angle
Identities (5.5)
Sum, Difference, Double Angle Identities (5.5)
Pencil/Paper
Use the sum or difference identity for tangent to find the exact
value of tan 285°.
Activity
Use the sum or difference identity for cosine to find the exact
value of cos 735°.
Performance
Verify that csc
3π
F
G
H2
Suggested Resources
IJ
K
+ A = − sec A is an identity.
Group Activity
If sin 2 = 2/3 , and 2 has its terminal side in the first quadrant,
find the exact value of each function:
a) sin 22
b) cos 22
c) tan 22
d) cos 42
28
Math Power 12 p.272 #1-27 odd,
Double Angle Worksheet (5.5)
Applications p.273 #37,39,40,41,
43(d),47
Enrichment
Applications Worksheet 5.5 at the
end of the unit.
Relating Graphs and Solutions Worksheet (5.1)
Graph each of the following using a graphing calculator. Inductively, find a formula for the exact
solution:
a) y = sin x ; y = 0
b y = sin2x ; y = 0
c) y = sin3x ; y = 0
d) y = sin x ; y = 1
e) y = sin 2x ; y = 1
f) y = sin 3x ; y = 1
g) y = cos x ; y = 0
h) y = cos 2x ; y = 0
i) y = cos 3x ; y = 0
j) y = cos x ; y = 1
k) y = cos 2x ; y = 1
l) y = cos 3x ; y = 1
29
Solutions for Worksheet (5.1)
Window settings
a) y = sin x ; y = 0
b y = sin2x ; y = 0
c) y = sin3x ; y = 0
nB;
nB/2
nB/3
d) y = sin x ; y = 1
e) y = sin 2x ; y = 1
f) y = sin 3x ; y = 1
B/2 + 2nB
B/4 + nB
B/6 + 2nB/3
90°, 450°
45°, 225°
30°, 150°, 270°
[0,4B,B/2],[!3,3,1]
[0,2B,B/4],[!3,3,1]
[0,2B,B/6],[!3,3,1]
g) y = cos x ; y = 0
h) y = cos 2x ; y = 0
i) y = cos 3x ; y = 0
B/2 + nB
B/4 + nB/2
B/6 + nB/3
90°, 270°
45°, 135°, 225°
30°, 90°, 150°
[0,4B,B/2],[!3,3,1]
[0,2B,B/4],[!3,3,1]
[0,2B,B/6],[!3,3,1]
j) y = cos x ; y = 1
k) y = cos 2x ; y = 1
l) y = cos 3x ; y = 1
2nB
nB
2nB/3
0°, 360°
0°, 180°, 360°
0°, 120°, 240°
[0,4B,B/2],[!3,3,1]
[0,2B,B/2],[!3,3,1]
[0,2B,B/3],[!3,3,1]
0°, 180°
0°, 90°, 180°
60°, 120°, 180°
30
[0,B,B],[!3,3,1]
[0,4B,B/2],[!3,3,1]
[0,4B,B/3],[!3,3,1]
31
Trig Equation Worksheet (5.2)
Solve for x:
1.
2 sin x cos x = 0,
2.
sin2 x + sin x = 0,
3.
cos2 x - 3 sin2 x = 1,
4.
tan x cos x - cos x = 0,
5.
sin 2x sin x = cos x, state the general solution
6.
2 cos2 x + 3 sin x - 3 = 0,
7.
cos2 x + 2 cos x - 3 = 0,
8.
cos2 x + 2 sin 2x = 0,
9.
tan2 x - 3 sec x + 3 = 0,
10.
cos x + tan x = 0,
32
Projectile Motion Activity (5.3)
OBJECT: Explore mathematically the time in the air, the maximum height achieved, and the
horizontal range of an object launched at various angles.
PROCEDURE: An object is launched at an initial speed of 30 m/s at the following angles. Use
Parametric Graphing to complete the following table.
2
tmax height
ttotal
y (height)
x (range)
100
200
300
400
450
500
600
700
800
900
Questions: At what angle will the object remain in the air for the longest time?
At what angle will the object have the largest horizontal range?
33
or animate
Press graph then trace
These last two screens show the time to reach the maximum height, the maximum height, the time to fall
back to Earth and the total distance travelled.
34
Trig Identity Worksheet (5.4)
Prove each of the following to be identities.
1
1.
1+
2.
sin A + tan A
= sin A
1 + sec A
3.
1 − sin 2 θ
= sin 2 θ
2
csc θ − 1
4.
sin A + cos A =
5.
tan 2 x − sin 2 x = sin 2 x tan 2 x
cos θ
=
tan 2 θ
sec θ − 1
1 + tan A
sec A
Determine if the following are identities.
6.
cot x + cos x =
cos x(1 + sin x)
sin x
7.
cot θ + cosθ =
2 cos θ
sin θ
8.
cot x + cos x = tan x + sin x
35
Double Angle Worksheet (5.5)
Prove each of the following to be identities:
11.
12.
14.
13.
15.
16.
17.
18.
19.
20.
21.
Simplify each of the following:
2.
1.
4.
3.
5.
36
Applications Worksheet (5.5)
1. Have you ever tried to tune in a radio station only to have it fade in and out or to have interference
from other channels disrupt your listening pleasure. This is called destructive or constructive
interference. What type of interference results when the following two signals are combined?
y = 20 sin(3t + 45°)
and y = 20 sin(3t + 120°)
TI-83 in deg mode; window dimensions [0,360,30],[!40,40,8] graph Y1 ,Y2 and Y3 = Y1 + Y2
2 In an electric circuit containing a capacitor, inductor and a resistor the voltage drop across the inductor
is given by VL = I0 TL cos(Tt + B/2), where I0 is the peak current, T is the frequency, L is the
inductance, and t is the time. Use the sum identity for cosine to express VL as a function of sin Tt.
3. Water fountains many times have water jets that shoot water into the air to create parabolic arcs. When
a stream of water is shot into the air at an angle of 2 with the horizontal, then water will travel a
horizontal distance of
D =
v2
v2
sin 2θ and reach a maximum height of H =
sin 2 θ
2g
g
where g is the acceleration due to gravity.
a)
H
as a function in simplest terms.
D
b) What is the ratio of the maximum height of the water to the horizontal distance it travels for an angle of
27°?
4. An AC circuit consists of a power supply and resistor. If the current in the circuit at time t is I0 sinTt,
then the power delivered to the resistor is P = I02 R sin2 Tt, where R is the resistance. Express the power
in terms of cos 2Tt.
5. The index of refraction for a medium through which light passes is the ratio of the velocity of light in a
vacuum to the velocity of light in the medium. For light passing through a prism the index of refraction is
1

sin  (α + β ) 
2

n=
β
sin
2
If $ = 60°, show that n =
where " is the deviation angle and $ is the angle of the apex of the prism .
3 sin
α
2
+ cos
α
2
.
Answers
37
1. destructive interference
2. VL = − I 0 ωL sin ωt
3.
H
1
=
tan θ
D
4
H
1
b)
=
tan 27 o ≈ 0.13
D
4
a)
4. P =
1 2
1
I 0 R − I 0 2 R cos 2ωt
2
2
5.
n =
=
=
=
=
1
sin[ (α + β)
2
β
sin
2
1
sin[ (α + 60o )
2
sin 30o
1
1
sin α cos 30o + cos α sin 30o
2
2
1
2
α
α
3
1
sin
2(
+ cos )
2
2
2
2
α
1
α
3 sin
+ cos
2
2
2
38