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Radians & Degrees

Angles in Standard Position Revisited

Trig. Functions & Graphs

Transformations

Trig. Identities

Trig. Eauations

Summary&Test

LESSON 7:Review & Summary

 

jdsmathnotes

 

 

 


Relation between Radians and degrees:

 

Text Box:

 

 

 

 

 

Example 1: Conversions from radian measure to degree measure

 

 

Solution:

 

Example 2: Conversions from degree measure to radian measure

 

Solution:

Text Box:
 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

 

 

 

 

 

 

 

 


Angles in Standard Position:

 

 

 

 

 

 

 

 

 

 

 

 

 

2. Find the exact value of :

 

 

 

 

 

 

Graphs of Trig. Functions

 

 

 

 

 

 

 

 

 

Combinations of Transformations -- y = a sin k(x d) + c and y = a cos k(x d) + c

Example 4: Sketch the graph of the basic function Use transformations to sketch .

Solution: For , construct a table of values of key points and plot the graph [blue]:

 

x0

0

90

180

270

360

y

0

1

0

-1

0

 

(x, y) ----------------------- (2x + 300, 3y + 1)

Using the mapping and the points from the table above we get points for the transformed graph

(x, y) ----------------------------- (2x + 300, 3y + 1)

(0, 0) ------------------------ (300, 1)

(900, 1) ---------------------- (2100, 4)

(1800, 0) --------------------- (3900, 1)

(2700 , -1) -------------------- (5700, -2)

(3600, 0) ---------------------- (7500, 1) [red graph at left]

In radians, the points (see table at top of page) would be:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Amplitude: |a| = 3

Domain of 1 period or cycle (between red dots on graph):

Range: The maximum value of y is 4 and the minimum value is 2

 

 

 

 

Text Box: In summary, to graph y = a sin [k(x  d)] + c from the graph of y = sin(x), follow these ideas:

	If a < 0, we have a reflection in the x-axis
	If k < 0, we have a reflection in the y-axis
	If  | a | < 1, we have a vertical compression , factor | a |
	If  | a | > 1, we have a vertical stretch, factor | a |
	
	If  | k | < 1, we have a horizontal stretch, factor 1/k
	If  | k | > 1, we have a horizontal compression, factor 1/k
	The value of d gives the horizontal translation (phase shift)
	The value of c gives the vertical translation (shift)
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Solving Trig. Equations:

 

 

 

 

 

 

 

 

 

 

 

 

 


Example:

 

Solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Text Box: Strategies for Proving Trig. Identities
1.	Start with the most complex side.  You may, however work on either side.
2.	Change any tan x expression to sin x and cos x using the quotient identity.
3.	Simplify algebraically using expanding or factoring where appropriate
4.	Use rules of fractions where needed  common denominators, multiplication and division rules for fractions.
5.	If sin2x or cos2x occurs, use the Pythagorean identities if they simplify the expression.
Basic Trigonometric Identities :

 

1. Quotient Identity:

 

2. Pythagorean Identities:

 

 

 

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