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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.
.
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:
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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:
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SUMMARY
Special Angles Table:
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1 |
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The Unit Circle for Special Angles