Ch 4 - Trigonometry
4.1 Radian and Degree Measure
4.2 Trigonometric Functions: The Unit Circle
4.3 Right Triangle Trigonometry
4.4 Trigonometric Functions of Any Angle
4.5 Graphs of Sine and Cosine Functions
4.6 Graphs of Other Trigonometric Functions
* (Graphs of Sec. and Csc. functions are Optional)
4.7 Inverse Trigonometric Functions
Objectives:
Today, we will discuss
Draw a Circle, Inside Your Notebook.
Write all angles (degrees & radian) and write all points
Unit Circle
Key numbers 1,2,and 3
Mini Lesson (Calculus)
4.2 Trigonometric Functions: The Unit Circle
Mini Lesson
Mini Lesson
How to remember Unit Circle?
Mini Lesson
How to remember Unit Circle?
The Unit Circle
It is a circle
with center and radius
whose equation is given by:
Wrapping Function
Each real number corresponds
to a point on the unit circle.
Then we can write:
1
x
x
Trigonometric Functions Definitions
Trigonometric Functions Definitions
Let be a real number and let
be the point on the unit circle corresponding to :
(f) "Cosecant function"
Example (4.2a)
If
Find:
t
t
t
( x , y)
(0,1)
Example (4.2a): Solution
Given:
Solution: we need the point on the unit
circle that corresponds to
1
2
(x, y)
3
Example (4.2b)
If
Find:
t
180
(x, y)
Example (4.2b): Solution
Given:
Solution: we need the point on the unit
circle that corresponds to
Mini Lesson
Unit Circle
Important Table to Remember (1)
Wrapping function of some important t’s:
t =
t =
t =
Important Table to Remember (2)
Continue…
t = OR
t =
t =
t =
t =
Example (4.2c)
Find the six trigonometric functions of:
(A)
Convert to Degree
Example (4.2c)
Unit Circle
Find the six trigonometric functions of:
(A)
Convert to Degree
?
135
Example (4.2c)
Unit Circle
Convert to Degree
Example (4.2c): Solution
(A) Solution:
(x, y)
1Angle
2 Point
3
Example (4.2c)
Find the six trigonometric functions of:
(B)
Negative
Example (4.2c)
Find the six trigonometric functions of:
(B)
Convert to Degree
?
-120
Negative
x , y
Negative
-120
1
2
3
4
Example (4.2c): Solution
(B) Solution:
Example (4.2d)
Find:
Example (4.2d): Solution
Required:
Example (4.2e)
Find:
If
find the image of
Example (4.2e): Solution
Required: Image of which is
Convert
Find the angle in degrees?
Decide on the quadrant?
Convert
Find the angle in degrees?
Decide on the quadrant?
Convert to Degree
?
390
30
30
360
390 or 30 degrees or pi/6 and First quadrant
Domain and Range of Sine Function
Domain and Range of Sine Function
The function is:
Domain:
Range: “without shifting”
Domain and Range of Cosine Function
The function is
Domain:
Range: “without shifting”
periodic functions each of period
Important Results
(a) "n is integer“
(b) "n is integer"
Remark:
Sine and Cosine functions are periodic
functions each of period
Convert to Degree
?
360
Example (4.2f)
(I) Evaluate:
Convert to Degree
390 - 360 = angle is 30
390
Example (4.2f): Solution
(I) Note:
Then use:
Example (4.2f)
(II) Evaluate:
Example (4.2f)
(II) Evaluate:
Convert to Degree
360 ? 720 ? 1080 ?
1215 - 1080 = angle is 135
1215
Example (4.2f): Solution
(II) Note:
Then use:
4.2 (continue)
Even and Odd Trigonometric Functions
Mini Lesson
Even and Odd Trigonometric Functions
Mini Lesson
Even and Odd Trigonometric Functions
Even Trigonometric Functions
(1) Cosine
(2) Secant
Odd Trigonometric Functions
(1) Sine
(2) Cosecant
(3) Tangent
(4) Cotangent
Even Trigonometric Functions
Odd Trigonometric Functions
(1) Cosine
(2) Secant
(1) Sine
(2) Cosecant
(3) Tangent
(4) Cotangent
Review: Even and Odd Functions “Sec. 1.5”
Example (4.2g)
Is the function
Even, Odd, or Neither?
Example (4.2g): Solution
Check:
Solution:
Since
then the function f is Even.
Replace x by - x
Even
Odd
Ready ?
Practice
Unit Circle Signs
Practice # 1
Find the following:
Practice # 2
Find:
Practice # 3
Is the function
Even, Odd, or Neither?
Even
Odd
Practice # 4
Find the exact value of:
Homework Problems�Section 4.2�page 297
7, 13, 21, 25,
29, 31, 39, 41,
43, 45
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