Download PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonomic Functions

PRE-CALCULUS: by Finney,Demana,Watts and Kennedy
Chapter 4: Trigonomic Functions
4.2: Trigonomic Functions of Acute Angles
What you'll Learn About


Right Triangle Trigonometry/ Two Famous Triangles
Evaluating Trig Functions with a calculator/Applications of right triangle trig
The six trigonometric functions
1|Page
Find the values of all six trigonometric functions.
2|Page
Assume that  is an acute angle in a right triangle satisfying the given
conditions. Evaluate the remaining trigonometric functions.
3|Page
A) sin  
4
9
C) tan 
4
9
D) cot  
E) csc 
10
7
F) sec 
B) cos  
2
9
2
9
4
3
45-45-90 Triangle
30-60-90 Triangle
4|Page
Evaluate using a calculator. Make sure your calculator is in the correct
mode. Give answers to 3 decimal places and then draw the triangle that
represents the situation.
A) sin 53
C) tan154
E) csc 220
5|Page
B) cos 
D) cot
F) sec
2
5

9
8
5
Solve the triangle for the variable shown.
Solve the triangle ABC for all of its unknown parts. Assume C is the
right angle.
  40 a = 10
6|Page
Solve the triangle ABC for all of its unknown parts. Assume C is the
right angle.
  62 a = 7
Example 6: From a point 340 feet away from the base of the Peachtree
Center Plaza in Atlanta, Georgia, the angle of elevation to the top of the
building is 65 . Find the height of the building.
7|Page
8|Page
PRE-CALCULUS: by Finney,Demana,Watts and Kennedy
Chapter 4: Trigonometric Functions
4.2: The circular functions
What you'll Learn About


Trig functions of any angle/Trig functions of real numbers
Periodic Functions/The Unit Circle
Point P is on the terminal side of angle  . Evaluate the six trigometric
functions for  .
A) (5, 4)
B) (-3, 4)
C) (-2, -5)
E) (0, -3)
9|Page
D) (-4, -1)
F) (3, 0)
Determine the sign (+ or -) of the given value without the use of a
calculator.
A) sin 53
C) tan154
E) csc 220
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B) cos 
D) cot
F) sec
2
5

9
8
5
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Evaluate without using a calculator by using ratios in a reference triangle.
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A) sin 120
B) cos 
C) tan
13
4
D) cot
E) csc
7
4
F) sec
2
3
 13
6
23
6
Find sine, cosine, and tangent for the given angle.
A)
90
B) 

2
D)  5
C) 6
Evaluate without using a calculator
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A) Find sin  and tan if cos  
3
and cot   0
4
B) Find sec and csc if cot  
6
and sin   0
5
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PRE-CALCULUS: by Finney,Demana,Watts and Kennedy
Chapter 4: Trigonometric Functions
4.3: Graphs of sine and cosine
What you'll Learn About


The basic waves revisited/Sinusoids and Transformations
Modeling
The graph of y = sin x
The graph of y = cos x
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Find the amplitude of the function and use the language of transformations to
describe how the graph of the function is related to the graph of y = sinx
A) y = 3sinx
B) y =
3
sin x
4
C) y = - 5sinx
Find the period of the function and use the language of transformations to describe
how the graph of the function is related to the graph of y = cosx
A) y = cos(2x)
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B)
y  cos
x
2
  3x 
C) y  cos

 4 
Graph 3 periods of the function without using your calculator.
A) y  3 sin
x
2
y  5 cos 2 x
Identify the maximum and minimum values and the zeros of the function in the
interval [2 ,2 ] . Use your understanding of transformations, not your calculator.
A) y = 4 sin x
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B) y  2 cos
x
3
Construct a sinusoid with the given amplitude and period that goes through the given
point.
A) Amp: 4, period 4 , point (0, 0)
B) Amp: 2.5, period

, point (2, 0)
5
State the Amplitude and period of the sinusoid, and relative to the basic function, the
phase shift and vertical translation.


A) y  3 sin x    2
4

C) y  5 sin 4x  6
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

B) y  2 sin 3x    4
4

Max  Min
2
M ax  M in
Vertical Shift 
2
period  p
Amp 
Horizontal Stretch/Sh rink
B
2
p
Example 7: Calculating the Ebb and Flow of Tides
One particular July 4th in Galveston, TX, high tide occurred at 9:36 am.
At that time the water at the end of the 61st Street Pier was 2.7 meters
deep. Low tide occurred at 3:48 p.m, at which time the water was only
2.1 meters deep. Assume that the depth of the water is a sinusoidal
function of time with a period of half a lunar day (about 12 hrs 24 min)
a) Model the depth, D, as a sinusoidal function of time, t, algebraically
then graph the function.
b) At what time on the 4th of July id the first low tide occur.
c) What was the approximate depth of the water at 6:00 am and at
3:00 pm?
d) What was the first time on July 4th when the water was 2.4 meters
deep?
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80) Temperature Data: The normal monthly Fahrenheit temperatures in
Helena, MT, are shown in the table below (month 1 = January)
Model the temperature T as a sinusoidal function of time using 20 as the
minimum value and 68 as the maximum value Support your answer
graphically by graphing your function with a scatter plot.
M
T
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1
20
2
26
3
35
4
44
5
53
6
61
7
68
8
67
9
56
10
45
11
31
12
21
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PRE-CALCULUS: by Finney,Demana,Watts and Kennedy
Chapter 4: Trigonometric Functions
4.5: Graphs of Tan/Cot/Sec/Csc
What you'll Learn About

The graphs of the other 4 trig functions
The graph of y = csc x
The graph of y = sec x
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The graph of y = tan x
The graph of y = cot x
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Describe the graph of the function in terms of a basic trigonometric function. Locate
the vertical asymptotes and graph 2 periods of the function.
A) y = 2tan(3x)
C) y = sec(4x)
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B) y = -cot(2x)
D)
y   csc 3x 
Describe the transformations required to obtain the graph of the given function form
a basic trigonometric graph.
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A)
y  5 tan x
C)
y  2 sec
4x
3
B)
y  3 cot  2x 
D)
y  4 csc 2x  3
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PRE-CALCULUS: by Finney,Demana,Watts and Kennedy
Chapter 4: Trigonometric Fucntions
4.7: Inverse Trigonometric Functions

What you'll Learn About
Inverse Trigonometric Functions and their Graphs
The graph of y  sin 1 x  arcsin x
The graph of y  cos 1 x  arccos x
The graph of y  tan1 x  arctan x
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The Unit Circle and Inverse Functions
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Find the exact value
A) cos -1
3
2
D) sin -1
 3
2
G) tan-1 1
J ) cos -1 0
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B) cos -1
 1
C ) cos -1  
 2 
1
2
E ) sin -1
 1 
F ) sin -1 

 2
1
2
 
H ) tan-1 3
K ) sin -1  1
 1 
I ) tan-1 

 3
L) tan-1 0
Use a calculator to find the approximate value in degrees. Draw the
triangle that represents the situation.
A) arccos (.456)
B) arcsin  .456
C ) arctan- 5.768
Use a calculator to find the approximate value in radians. Draw the
triangle that represents the situation.
A) arcsin (.456)
C ) arctan- 5.768
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B) arccos  .456
Find the exact value without a calculator.


A) sin cos 1 1 2
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

B) cos tan1 0

 2 

C ) tan sin 1 


2



D) sin tan1  3
   
E ) cos -1  sin  
  4 
   
F ) sin -1  cos  
  6 



32 | P a g e
PRE-CALCULUS: by Finney,Demana,Watts and Kennedy
Solving Trigonometric Equations
What you'll Learn About
Solve each trigonometric equation for  on the interval 0,2 .
A) sin  
2
2
C) sin   1
E) cos 2 
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B) cos  
1
2
D) cos   0
1
2
F) sin 3 
1
2

G) cos 
3
I) sin   .4
34 | P a g e
3
2
 
H) tan    1
2
3
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A)
2 sin   1  0
C)
4 cos 2   1  0
E)
2 cos 2   cos   0
B)
3 sec  2  0
D) 3 tan2   1tan2   3  0
F) 2 sin  cos   cos 
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PRE-CALCULUS: by Finney,Demana,Watts and Kennedy
Chapter 5: Analytic Trigonometry
5.1 Fundamental Identities
Reciprocal Identity Example
csc

4

1
sin

4
Even/Odd Identity Example
cos
37 | P a g e

  
 cos

4
 4 
Quotient Identity Example
tan

4

sin
cos

4

4
Cofunction Identity Example
sin90  60  cos(60)
Equation of Unit
Circle
x2  y 2  1
2
2
Use trig ratios to prove that cos   sin   1
is the equation of the unit circle
Use basic identities to simplify the expression to a different trig function
or a product of two trig functions
1  sin 2 
10. cot x tan x
A.
cos 
B. sin x  sin x
3
38 | P a g e
C.
sin 2 x  cot 2 x  cos 2 x
csc x
Simplify the expression to either 1 or -1
17. sin x csc( x)
19.


cot  x cot   x 
2

2
2
21. sin  x   cos ( x)
Simplify the expression to either a constant or a basic trig function.


cot   x  sec x
1  cot x
2

A)
B)
2
1  tan x
sec x

2
2
2
2
C) tan x  cot x  sec x  csc x
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
Use the basic identities to change the expression to one involving only
sines and cosines. Then simplify to a basic trig function.


28) sin   tan cos   cos   
2

30)
31.
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sec y  tan y sec y  tan y 
sec y
tan x
tan x

2
csc x sec 2 x
Combine the fractions and simplify to a multiple of a power of a basic
trig function
1
1

A)
1  cos x
1  cos x
35.
sin x
sin x

2
cot x
cos 2 x
Write each expression in factored form as an algebraic expression of a
single trig function
A) sin 2 x  2 sin x  1
C) sin x  2 cos 2 x  1
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B) 1  2 cos x  cos 2 x
45. 4 tan2 x 
4
 sin x csc x
cot x
Write each expression as an algebraic expression of a single
trigonometric function
A)
1  cos 2 x
1  cos x
cos 2 x
C)
1  sin x
cot 2   1
1  cot x
B)
D)
cot 2 x
1  csc x
Find all solutions to the equation in the interval [0, 2π]
A) 2cosxsinx + sinx = 0
C) cos x tan2 x  cos x
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B)
2 cot x sin x  cot x  0
D) cot 2 x  3
Find all solutions to the equation in the interval [0, 2π]
A) 4 sin 2 x  4 sin x  1  0
B) cos 2 x  2 cos x  0
C) 3 cos x  2 sin 2 x
D)
2 cos 2 x  3 cos x  2
Find all solutions to the equation in the interval [0, 2π]
A) cosx = .53
43 | P a g e
B) sinx = .75
C) cos2x=.9
44 | P a g e
PRE-CALCULUS: by Finney,Demana,Watts and Kennedy
Chapter 5: Analytic Trigonometry
5.2 Proving Trig Identities
What you'll Learn About
12. sinxcotx  cosxtanx   cos x  sin 2 x
14. cosx  sinx   1  2 sin xcos x
2
18.
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sec2  1
sin

sin 
1  sin 2 
20.
1
1

 2 csc2 x
1 - cosx 1  cos x
22. sin 2  cos 2  1  2cos 2
26.
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secx  1
sinx

tanx
1 - cosx
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30.
tan 2  sin 2   tan2  sin 2 
35.
tanx
secx  1

secx - 1
tanx
37.
sinx - cosx
2 sin 2 x  1

sinx  cosx 1  2sinxcosx
48 | P a g e
49 | P a g e