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Radians & Degrees

Trig. Definitions using {x, y, r}

Trig. Definitions using Unit Circle

Trig. Functions & Graphs

Transformations

Trig. Identities

Trig. Eauations

Summary&Test

 

jdsmathnotes

 


 UNIT 7 : TRIGONOMETRIC FUNCTIONS

 LESSON 8 : TRIGONOMETRIC FUNCTIONS REVIEW

 

 

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 using {x, y, r}:

 

Definition: An angle is in standard position if it has its vertex at the origin and initial arm along the positive x-axis. The terminal arm is found by rotating

the initial arm about the origin to a terminal position in one of the 4 quadrants. The rotation is positive if it is in the counter clockwise direction and negative

if in the clockwise direction.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Angles in Standard Position using the Unit Circle:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Definition: Coterminal angles have the same initial arm and same terminal arm. They can be found by adding or subtracting 3600 from the given angle.

 

 

 

 

 

Summary:

 

 

 

Special Angles Table:

 

1

 

The Unit Circle for Special Angles

 

 

 

 

 

 

 

 

 

 

 

 

 

Example :

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Trigonometric Functions of Real Numbers Using the CAST RULE & Special Angles Table :

 

The CAST rule:

 

The CAST RULE is a memory aid which tells us the sign of the trig ratios in the various quadrants.

In the first quadrant ALL are positive. This is denoted using the letter A.

In the second quadrant, SINE is positive. This is denoted by the letter S.

[ the other two ratios are negative ]

In the third quadrant, TANGENT is positive. This is denoted by the letter T.

[ the other two ratios are negative ]

In the fourth quadrant, COSINE is positive. This is denoted by the letter C.

[ the other two ratios are negative ].

 

 

 

 

Special Angles Table:

 

1

 

 

 

 

 

 

Special Angles Table:

 

1

 

 

 

 

 

 

 

 

 

 

 

 

Graphs of Trig. Functions

 

 

 

 

 

 

 

 

 

**See lesson 3 for the reciprocal graphs

 

 

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

 

 

 

x0

0

90

180

270

360

y

0

1

0

-1

0

 


 

 


 

 

 

 

 

 

 

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:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Basic Trigonometric Identities :

 

 

 

 

 

 

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 or cot x expression to sin x and cos x using the quotient identities.
3.	Change any sec x or csc x expression to sin x and cos x using the reciprocal identities.
4.	Simplify algebraically using expanding or factoring where appropriate
5.	Use rules of fractions where needed  common denominators, multiplication and division rules for fractions.
6.	If sin2x or cos2x etc.  occurs, use the Pythagorean identities if they simplify the expression.
This should be done if the number 1 occurs with the squared expression.
 

 

 

 

 

 

 

 

 

 

 

 

 

 


Example :

 

Proof:

 

 

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