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4.01 Angles and Their Measure
Keywords
Angle
Definitions/Examples
Initial side:
Terminal Side:
Vertex:
Copy 3
images from
lesson
Standard Position:
Positive Angle:
Coterminal Angles:
Negative Angle:
Radian
Measures
We measure an angle by stating the amount of rotation from the initial side
to the terminal side.
Radian:
Acute Angles:
Obtuse Angles:
Degree
Measure
Convert
between
Degrees and
Radians
To convert from degrees to radians:
To convert from radians to degrees:
Test your
skills
Application:
Find arc
length
Formula for arclength:
If the radius of a circle is 4", what is the length of the arc measured by Pi
symbol/2 radians?
4.02 Trigonometric Functions of Acute Angles
Keywords
Right Triangle
Definitions/examples
Hypotenuse:
Opposite side:
Adjacent Side:
Six Trigonometric Functions
http://youtu.be/5tp74g4N8EY
Test your skills
Determine the six trigonometric functions for the
following right triangles. Choose the Check Your
Answers link below to view the solutions to these
problems.
adjacent side = 6 and opposite side = 8
adjacent side = 4 and opposite side = 4
hypotenuse = 2 and adjacent side = 1
Special Trigonometric
Values you should
Memorize
Trigonometric Identities
Reciprocal Identities:
Quotient Identities
Pythagorean Identities
Test your skills
Answer each of the following. Choose the Check
Your Answers link below to view the solutions to
these problems.
a. In a right angle, an acute has cos = .6.
Determine sin and tan.
b. A visitor to the Washington Monument is
curious about the height of the monument.
He walks away from the monument until his
line of sight to the top of the monument is
exactly 60°. He has taken 107 steps to this
point and each step is 3 feet long. How high
in feet is the Washington Monument?
4.03 Trigonometric Functions: The Unit Circle
Keywords
The Unit Circle
Definitions/Examples
Definition of Trigonometric
Functions
Let x be a real number and (x,y) the point on the unit
circle corresponding to t.
Sin(t)=
Cos(t)=
Tan (t)=
Csc (t)=
Sec(t)=
Test your
Skills
Cot(t)=
Determine the values of the six trigonometric functions
for the following.
a. t = 4π /3
b. t=3π /4
Domain and Period of Sine
and Cosine
Definition of a Periodic
Function
Test your skills
Determine the values of the six trigonometric functions
for the following.
a. t=15π/6
b. t= 11π/2
4.04 Trigonometric Functions of Any Angle
Keywords
Definitions of
Trigonometric
Functions of any
Angle
Definitions/examples
Let be an angle in standard position with (x,y) a point on the
terminal side of and
r=
Sin=
Cos=
Tan=
Cot=
Sec=
Csc=
Example: If an angle  has a terminating side which contains
point (1, √3), what are the values of the sine, cosine, and
tangent of ?
Definition/Examples:
Trigonometric
Functions
of
Real Numbers
Test your skills:
Answer each of the following.
A. For an angle  with the point (-3, -4) on its terminating side, what are
the values of the sine, cosine, and tangent?
B. For an angle  with the point (-1,-1) on its terminating side, what are
the values of the sine, cosine, and tangent?
C. If tan = -1, and the sintheta > 0, what is the value of cos?
D. Determine the values of cos Pi, sin(3Pi/2), and tan(2Pi).
*** Check your answers within the lesson *****
Definition:
Reference
Angles
If the angle  is in the 2nd quadrant, then the reference angle
is _______________
For an angle  in the 3rd quadrant, the reference angle is
_________________
And for an angle 
_______________
in the 4th quadrant, the reference angle is
Test Your Skills
Determine the reference angle for the following.
a. 100°
b. 225°
c. -100°
*** Check your answers within the lesson *****
Definitions/Examples:
Trigonometric
Functions of Real
Numbers
ASTC
A
S
T
C
4.05a Graphs of Sine and Cosine Functions
Keywords:
Definitions/Examples
Basic Sine
and Cosine
Curves
Sine Curve:
Key points on Maximum and Minimum points:
a Sine or
Cosine Curve
X-axis intercepts
Y-axis intercept
The Standard
Form of the
Equations for
Sine and
Cosine
Sine and Cosine curves can be expressed in the following standard
equations:
Amplitude
Period
Phase Shift
Vertical Shift
Summary
Test your skills:
Determine the amplitude, period, and left and right endpoints for the following.
***Check your answers within the lesson*****
Graphing by
Hand
How to graph by hand:
Test your skills
Graph each of the following trigonometric curves by hand. Find the key points to
help you plot an accurate curve. You can check your answers by graphing the
functions with your calculator (be sure that your calculator is in radian mode).
Use the trace key to check the values of the intercepts and the maximum and
minimum values.
Test your skills
***Check your answers within the book***
4.06 Graphs of other Trigonometric Functions
Keywords
The Standard Form of
the Tangent Function
Definitions/ examples
Definitions/ examples/notes
About the Graph of the
Tangent Function
Definitions/ examples/notes
Summary of How to
Graph the Tangent by
Hand
1.
2.
3.
4.
5.
Graph y = 2 tan (3x + pi)
1.
2.
3.
4.
5.
Test your skills
For each of the following tangent curves, list the new period, the phase shift,
equation of the vertical asymptotes, and the intercepts.
Graph of the Cotangent
Function
Definitions/ examples/notes
Graphs of the
Reciprocal Functions
Definitions/ examples/notes
Graphing Review
Definitions/ examples/notes
4.07 Inverse Trigonometric Functions
Keywords
Definitions/Examples/Notes
Inverse Sine
Function
Definitions/Examples/Notes:
For this domain, the following properties exist:
1.
2.
3.
Definition of
Inverse Sine
Function
The inverse sine function is defined by
Test your skills:
Evaluate each of the following without using a calculator.
***Check your answers within the lesson***
Other Inverse
Trigonometric
Functions
Definitions of the
Inverse
Trigonometric
Functions
Domain
Range
Test your skills
Evaluate each of the following without using a calculator. Remember to pay attention to
the range of the inverse function, and give your angle in the correct quadrant.
a. arccos(1)
b. cos-1(0)
c. arccos(0)
d. arctan(-1)
e. tan-1(√3)
**Check your answers within the lesson***
Compositions of
Functions
Definitions/Examples/Notes:
Example:
Test your skills
Find the value of each of the following. You can check your answers by typing the
expression into your calculator.
1. sin ( sin-1 (1))
2. cos (sin -1 (-0.5))
3. tan-1 (sin /2)
4.08 Solving Problems with Trigonometry
Keywords
Definitions/Examples/Notes
Applications
Involving Triangles
Definitions/Examples/Notes :
Simple harmonic
motion
Definitions/Examples/Notes:
Harmonic Motion
A point that moves on a coordinate line is said to be in simple
harmonic motion if its distance d from the origin at time t is
given by either