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