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Math Analysis
Chapter 4 Notes: Trigonometric Functions
Day #1: Section 4-1: Angles and Radian Measure; Section 4-2 Trigonometric Functions
After completing section 4-1 you should be able to do the following:
1. Use degree measure
2. Use radian measure
3. Convert between degrees and radians
4. Draw angles in standard position
5. Find coterminal angles
6. Find the length of a circular arc
Angles
An angle is formed by two rays that have a common endpoint (vertex). One ray is called the initial side and the other the
terminal side. If the angle is in standard position the vertex is at the origin of a rectangular coordinate systerm and the
initial side is always along the positive x-axis.
Standard Position of an Angle:
Terminal Side
•
Initial Side
This angle θ is a positive angle.
The direction of rotation from the
initial side to the terminal side is
counter-clockwise.
Positive and Negative Angles
 Positive Angle: When a ray is
rotated from the initial side
counter-clockwise, the angle
measure is positive.
 Negative Angle: When a ray is
rotated from the initial side
clockwise, the angle measure is
negative.
•
This angle θ is a negative angle.
The direction of rotation from the
initial side to the terminal side is
clockwise.
Degree Measure of an Angle
Angles are measured by determining the amount of rotation from the initial side to the terminal side. One way to measure
angles is in degrees, symbolized by a small, raised circle 0. A complete rotation around a circle is considered 360 0, therefore
1
10 =
of a complete rotation around a circle.
360
Classifying angles by there degree measurement
•
00 < θ < 900
•
•
θ = 900
900 < θ < 1800
•
θ = 1800
Practice: In 1-4, Draw the given angle in standard position. State the quadrant the terminal side is in.
Math Analysis Notes Mr. Hayden
1
1. 450
2. 2250
4. −600
3. 2700
Radian Measure of an Angle
Another way to measure angles is in radians. One radian is the measure of the central angle of a circle that intercepts an arc
length equal in length to the radius of the circle:
•
θ = 1 radian =
arc length r
  1 radian = 1
radius
r
A radian is the ratio of the arc length (S) intercepted by two radii. A radian is a unit-less angle measurement.
Conversion between Degrees and Radians
1.
To convert degrees to radians, multiply degrees by
2.
To convert radians to degrees, multiply radians by

1800
1800

.
.
Practice: In 5-7, Convert each angle in degrees to radians. Do not use a calculator.
5. 600
6. 2700
7. −3000
Practice: in 8-10, Convert each angle in radians to degrees. Do not use a calculator.
8.

4
Math Analysis Notes Mr. Hayden
9. 
4
3
10. 1
2
The following are very important equivalent forms of radian and degree measures. The sooner you realize they are the same
the better you will do in the next three chapters on Trigonometry.
300 =

6
600 =

3
900 =

2
1800 = π
2700 =
3
2
3600 = 2π
Coterminal Angles
Two angles when drawn in standard form are said to be coterminal angles if they have the same terminal side.
The angles 2250, 5850 and
−1350 are said to be
conterminal because they all
have the same terminal side
when drawn in standard
position.
•
To find Coterminal Angles:
 In Degree Measure: add multiples of 3600 or subtract multiples 3600 to the angle given
 In Radian Measure: add multiples of 2π or subtract multiples of 2π to the angle given
Practice: In 11-14, Find a positive angle less than 3600 or 2π that is coterminal with the given angle.
12. −1350
11. 4000
13.
22
3
14. 
17
6
The Length of a Circular Arc
Let r be the radius of a circle and θ the
nonnegative radian measure of a
central angle of the circle. The length of
the arc intercepted by the central angle
is:
•
S = rθ
Practice: In 15-17, Find the length of the arc on a circle of radius r intercepted by a central angle θ.
15. r = 10 inches, θ = 450
16. r = 5 feet, θ = 900
Math Analysis Notes Mr. Hayden
17. r = 6 yards, θ =
2
3
3
4.2
The word trigonometry means measurement of triangles. You need to rememorize your basic right triangles:
450-450-900
300-600-900
The six trigonometric functions are:
Name
sine
cosine
tangent
Abbreviation
sin
cos
tan
Name
cosecant
secant
cotangent
Abbreviation
csc
sec
cot
The six trigonometric functions are defined as:
opposite
hypotenuse
csc  
cos  
adjacent
hypotenuse
sec  
hypotenuse
adjacent
tan  
opposite
adjacent
cot  
adjacent
opposite
sin  
hypotenuse
opposite
Use SOH-CAH-TOA to
remember the 1st 3 trig
functions.
Steps to find the trigonometric values of angle:
1. Draw angle in standard position
2. Draw an altitude from the x-axis to the terminal side of the triangle
3. Determine type of triangle created and use the above definitions to find the trigonometric value.
Practice: In 18-21, Find the six trigonometric values of the given angle.
18. 1350
19. −600
Math Analysis Notes Mr. Hayden
20.
5
6
21.
5
3
4
Worksheet on Evaluating Trigonometric Functions
Math Analysis
(Part of HW #1)
Please do work on separate sheet of paper.
In 1-2, Draw the angle in standard position and find the six trigonometric value of the given angle.

1. θ = 2100
2. θ  
3
In 3-10, Draw the angle in standard position and then evaluate the trigonometric function.
3. sec1350
4. tan2400
5. sin(–1500) 6. csc(–4200)
7. cos
7
4
11
 8 
 3 
8. cot  
 9. tan  
 10. sec
6
 3 
 4 
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Day #2: Section 4-2: Trigonometric Function; Section 4-3 Right Triangle Trigonometry; Sections 4-4
Trigonometric Functions of Any Angle
Unit Circle
A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit
circle is x2 + y2 = 1.
Definitions of the Trigonometric Functions in Terms of a Unit Circle
If t is a real number and P = (x, y) is a point on the unit circle that corresponds to t, then
1
csct 
sin t  y
y
cos t  x
tan t 
y
x
1
x
x
cot t 
y
sec t 
Practice: In 1-2, Given a point P(x,y) is shown on the unit circle corresponding to a real number t. Find the values of
the six trigonometric functions at t.
1.
Math Analysis Notes Mr. Hayden
2.
6
The Unit Circle:
Practice: In 3-7, use the unit circle to evaluate the trigonometric function.
3. sin

4. cos
6
5
6
5. tan 
6. csc
7
6
7. sec
4
3
Trig Identities
Reciprocal Identities
1
sin  
csc 
cos  
1
sec 
tan  
1
cot 
1
sin 
sec  
1
cos 
cot  
1
tan 
Quotient Identities
sin 
tan  
cos 
cot  
cos 
tan 
csc  
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Pythagorean Identities
sin2   cos2   1
1  tan2   sec2 
1  cot 2   csc2 
Practice: In 8-13, Use the trigonometric identities to evaluate or simplify.
8. cos  
11. sin 2

3
1
find cos    ?
2
 cos 2

9. sin    
12. sec2
3

4
2
find sin  ?
3
 tan 2
10. sin(0.2) csc(0.2)

13. Find cos  given sin  
4
6
7
4-3 Right Triangle Trigonometry
Practice: In 14-15, find the six trigonometric functions of the given right triangle.
14.
15.
Practice: In 16-19, Draw the angle in standard position then evaluate the trigonometric function
16. cos300
17. tan(−600)
Math Analysis Notes Mr. Hayden
18. csc2250
19. cot2100
8
Practice: In 20-21, Find the length of the missing side of the triangle.
20.
21.
Practice: In 22, a point is given on the terminal side of an angle θ is given. Find the six trig functions of θ.
22. (−4, 3)
Day #3: Review of Sections 4-1 to 4-4 and Chapter 4 Quiz
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Day 4: Section 4-3 Applications of right triangles, Section 4-5 Graph of Sine and Cosine Functions
After completing today notes you should be able to do the following
 Solve problems involving angle of elevation
 Solve problems involving angle of depression
 Graph sine equations
 Graph cosine equations
4-3: Applications
Many applications of right triangle trigonometry involve the angle made with an imaginary horizontal line. An angle formed
by a horizontal line and the line of sight to an object above the horizontal line is called the angle of elevation.
Line of sight above observer
Observer located
here
Angle of
Elevation
Horizontal Line
The angle formed by a horizontal line and the line of sight to an object that is below the horizontal line is called the angle of
depression.
Observer located
here
Horizontal Line
Angle of
Depression
Line of sight below observer
Practice: In 1-2, Solve each problem.
1. A flagpole is 14 meters tall casts a shadow 10 meters long. Find the angle of elevation of the sun to the nearest degree.
Math Analysis Notes Mr. Hayden
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2. On a cliff 250 feet above the sea an observer sights a ship in the water. If the angle of depression is measured to be 15 0,
how far is the ship from the cliff?
4-5: Graphs of Sine and Cosine Functions.
Values of (x, y) on the graph y  sin x :
Values of (x, y) on the graph y  cos x
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11
The sine or cosine graph can vary according to the different values of a, b, c, and d.
y  a sin( bx  c)  d
y  a cos(bx  c)  d
where
a = amplitude: vertical distance from starting point to the next point x-value ¼ period.
b – helps find period: Trigonometric functions are periodic, which means that the graph has a repeating pattern that
continues indefinitely. The shortest repeating portion is called a cycle. The horizontal length of each cycle is called the
period. To find the period of a sine or a cosine function use:
period =
2
b
Interval length: How far horizontally to go to find next exact value of the graph. To find interval length use:
Interval length =
period
4
c ─ helps find phase shift. The phase shift is a horizontal shift that states the starting value of the sine graph. To find the
phase shift set bx – c = 0 and solve for x. The value of x is the phase shift.
d = vertical shift:
Practice: In 1-4 Graph the Sine or Cosine function. You must graph two full periods for full credit.
1. y  3 sin( 2 x   )  1
 x

 3   3
 2

2. y  2 sin 
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3. y 
1
1
cos x
2
3


4. y  2 cos 3 x 

3
2
Practice: In 5-6, find both the sine and cosine equation for each graph. (pg 518 #61 and 63)
5.
Math Analysis Notes Mr. Hayden
6.
13
Day 5: Section 4-6 Graph of other Trigonometric Functions
After completing today notes you should be able to do the following
 Graph tangent function
 Graph cotangent function
 Graph cosecant function
 Graph secant function
Values of (x, y) on the graph
y  tan x :
The tangent graph can vary according to the different values of a, b, c, and d.
y  a tan( bx  c)  d
where
a = vertical distance from starting point to the next point.
b – helps find period: Trigonometric functions are periodic, which means that its graph has a repeating pattern that
continues indefinitely. The shortest repeating portion is called a cycle. The horizontal length of each cycle is called the
period. To find the period of a tangent function use:

b

bx  c 
2
period =
Vertical Asymptotes: can be found by solving both:
and bx  c 

2
Interval length: How far horizontally to go to find next exact value of the graph. To find interval length use:
Interval length =
period
4
c ─ helps find phase shift. The phase shift is a horizontal shift that states the starting value of the sine graph. To find the
phase shift set bx – c = 0 and solve for x. The value of x is the phase shift.
d = vertical shift:
Math Analysis Notes Mr. Hayden
14
Practice: In 1-2, Graph the tangent function.
1.
y  3 tan 2x  4 
2. y  
1
tan 2x 
2
Math Analysis Notes Mr. Hayden
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Graph of Cotangent
y  cot x
In 3-4, Graph the Cotangent function.
3. y  2 cot x
4.
y   cot2x   
Math Analysis Notes Mr. Hayden
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Graph of Cosecant Function
y  csc x
In 5, Graph the Cosecant Function
5.
y  csc2x  4   2
Graph of Secant Function
y  sec x
In 6, Graph the Secant Function
6.
y  secx  2 
Math Analysis Notes Mr. Hayden
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Day 6: Section 4-7 Inverse Trigonometric Functions
After completing today notes you should be able to do the following
 Understand and use the inverse sine function
 Understand and use the inverse cosine function
 Understand and use the inverse tangent function
 Use a calculator to evaluate inverse trigonometric functions
Remember that in order for a function to have an inverse it must past the horizontal line test.
The inverse sine function has two notations are commonly used to denote the inverse sine function:
y  sin 1 x
or
y  arcsin x
The inverse sine function is restricted to the interval: 

2
x

2
. Which means the inverse sine function is only
defined in the 1st and 4th quadrants.
When you are finding the value of the inverse sine function you are finding an angle value.
Practice: In 1-2, Evaluate the given function.
1. sin
1
 3


 2 


2. sin
1

2


 2 


The inverse cosine function has two notations are commonly used to denote the inverse sine function:
y  cos 1 x
or
y  arccos x
The inverse sine function is restricted to the interval:
defined in the 1st and 2nd quadrants.
0  x   . Which means the inverse sine function is only
When you are finding the value of the inverse cosine function you are finding an angle value.
Practice: In 1-2, Evaluate the given function.
 3

 2 


1
1. cos 
Math Analysis Notes Mr. Hayden
1
 1

 2
2. cos  
18
The inverse tangent function has two notations are commonly used to denote the inverse sine function:
y  tan 1 x
or
y  arctan x
The inverse sine function is restricted to the interval: 

2
x

2
. Which means the inverse sine function is only
defined in the 1st and 4th quadrants.
When you are finding the value of the inverse tangent function you are finding an angle value.
Practice: In 1-2, Evaluate the given function.



1
1. tan  
3

3 
2.
tan 1  1
Practice: In 1-6, Evaluate the given function.
1. cos sin 1 4 
2. sec sin 1   1  

4 
3. tan  cos 1   1  

3 
4. cos tan 1 2 


5



Math Analysis Notes Mr. Hayden



3
19
5. tan cos 1 x 
6. sec sin 1





x 4
x
2
Using a calculator to evaluate inverse trigonometric functions.
You need to verify what quadrant the angle lands in. Your calculator will only give you reference angles.
Practice: In 1-4, use a calculator to find the value of each expression rounded to two decimal places.
1.
sin 1 (0.32)
 5

 7 


1
3. cos 
Math Analysis Notes Mr. Hayden
2. cos
4.
1
3
8
sec 1 (0.25)
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Day 7: Section 4-8 Solving Right Triangles
After completing today notes you should be able to do the following
 Solve Right Triangles
A right triangle has 6 parts, 3 angles and 3 sides. To solve a right triangle means to find the length of all 3 sides and find the
measure of all 3 angles.
Practice: 1. Solve the right triangle.
C
b
A
Given B = 23.50 and c = 10
a
c
B
Practice: 2. Find x.
Math Analysis Notes Mr. Hayden
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Chapter 4 Review Worksheet
Please do all work on a separate piece of paper. Show all work.
In 1-2, Evaluate the six trigonometric functions of the given angle θ.
1.
2.
θ
15
9
12
θ
10
In 3-6, Draw an angle with the given measure in standard position.
3. 1000
4. −450
5.
5
9
6.
12
5
In 7-10, Find one positive angle and one negative angle that is coterminal with the given angle.
7. 2250
8. 600
9.
15
2
10.
16
5
In 11-18, Rewrite each degree measure in radians and each radian measure in degrees.
11. 1350
15.
7
12
12. 400
16.
5
6
13. 2600
17.
14. 2150
2
3
18.

6
In 19-21, Find the arc length with the given radius r and centeral angle θ.
19. r = 4 in, θ = 600
20. r = 5 m, θ = 2400
21. r = 12 cm, θ =
5
3
In 22-23, Solve ΔABC using the diagram at the right and the given measurements.
22. B = 400, a = 14
23. A = 350, a = 12
A
C
Math Analysis Notes Mr. Hayden
c
b
a
B
22
In 24-25, Use the given point on the terminal side of an angle θ in standard position. Evaluate the six
trigonometric functions of θ.
24. (−4, 5)
25. (−5, −8)
In 26-33, Draw the angle in standard position then evaluate the function without using a calculator.
27. sin(−600)
26. tan 1350
28. cos 2100
29. sec(−3150)
7
2
7
31. csc
32. tan
33. cos  6900 
6
3
3
In 34-37, Use a calculator to evaluate the function. Round the result to four decimal places.
30. cot
34. sin 180
35. sec 4
36. cot (−6.7)
37. csc2420
In 38-39, Solve each problem. Round measurements of lengths to the nearest tenth. Draw a picture,
set up an equation, and then solve.
38. Nancy shines a light from a window of Rocky Rococco’s beachside mansion on a cliff 250 feet
above the water level. Nick Danger, 10 feet above the water level, is on a ship off –shore find the angle
of elevation of the light is 50. Find the slant distance from the ship to Rococco’s mansion.
39. An airplane is directly above a beacon that is 10,000 feet from an airport control tower. The angle
of depression from the plane to the base of the control tower is 60. How high above the beacon is the
plane?
In 40-45, Determine the amplitude, period, interval lengths, phase shift, and vertical shift. Then graph
the function. Label the axes. Graph 2 full periods.
x 
40. y  5sin 
2
 4 


41. y  2sin  2 x    2
2

42. y  4 cos x  
43. y 
44. y  2 tan  x


45. y  tan  x   
2

46. y  3 sec(
x
4


4
) 1
47. y 
1
cos  4 x  2   3
2
1
csc2 x   3
2
48. y  cot x
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