Keywords Definitions/Examples
Angle 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 MeasureConvert between Degrees and Radians
To convert from degrees to radians:
To convert from radians to degrees:
4.01 Angles and Their Measure
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?
Keywords Definitions/examples
Right Triangle Hypotenuse:
Opposite side:
Adjacent Side:
4.02 Trigonometric Functions of Acute Angles
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 = 1Special 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?
Keywords Definitions/Examples
The Unit Circle
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)=
Cot(t)=Test your Skills
Determine the values of the six trigonometric functions for the following.
a. t = 4π /3
b. t=3π /4Domain 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
4.03 Trigonometric Functions: The Unit Circle
b. t= 11π/2
4.04 Trigonometric Functions of Any Angle
Keywords Definitions/examples Definitions of Trigonometric Functions of any Angle
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 ?
Trigonometric Functions of Real Numbers
Definition/Examples:
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 *****Reference Angles
Definition:
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 *****Trigonometric Functions of Real Numbers
Definitions/Examples:
A S T C 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 a Sine or Cosine Curve
Maximum and Minimum points:
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 Definitions/ examples The Standard Form of the Tangent Function
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 skillsFor 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