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Radians & Degrees

Trig. Definitions using {x, y, r}

Trig. Definitions using Unit Circle

Trig. Functions & Graphs

Transformations

Trig. Identities

Trig. Equations

Summary&Test

 

jdsmathnotes

 


 UNIT 7 : TRIGONOMETRIC FUNCTIONS

 LESSON 3: ANGLES IN STANDARD POSITION AND THE UNIT CIRCLE

 

Angles in Standard Position:

 

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.

 

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Example 1:

 

 

 

 

 

 

 

 

 

 

 

 

 

Note that all three ratios are positive for a first quadrant angle.

 

 

 

 

 

 

 

 

 

Note that sine is positive and cosine and tangent are negative for a second quadrant angle.

 

 

x = -3

y = -2

 

 

 

 

 

 

Note that tan is positive and sine and cosine are negative for a third quadrant angle.

 

 

 

 

 

 

Note that cos is positive and sine and tangent are negative for a fourth quadrant angle.

 

 

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 ].

 

 

 

 

 

 

 

 

 

 

 

 

 

The primary trigonometric functions for the special angles may be summarized in the following table.

This table should be memorized to complete this unit successfully.

 

Special Angles Table:

 

 

1

 

 

 

Trig. Functions of any Angles in Standard Position:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Quadrantal Angles:

 

Definition: Quadrantal angles are angles whose terminal arms end at one of the coordinate axis.

Examples : 900, 1800, 2700, 3600, -900, etc.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

Special Angles Table:

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SUMMARY

 

 

 

 

Special Angles Table:

 

1

 

The Unit Circle for Special Angles

 

 

 

 

 

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