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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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1 |
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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