Trigonometry - PP2
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Transcript of Trigonometry - PP2
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PP2 of series
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STANDARD POSITION
Let P(x,y) represent
a point which moves
around a circle with
radius r and center (0,0)
The measure of theangle may be in
degrees or in radians.
A (r,0)X
y
Initial arm
terminal arm
O
P(x,y)
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STANDARD POSITION
Let P(x,y) represent
a point which moves
around a circle with
radius r and center (0,0)
The measure of theangle may be in
degrees or in radians.
A (r,0)
X
y
P(x,y)
Initial arm
terminal arm
O
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IF > 0, THE ROTATION IS
COUNTERCLOCKWISE
A (r,0)
X
y
O
P(x,y)
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IF < 0, THE ROTATION IS CLOCKWISE
A (r,0)
X
y
P(x,y)
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A (r,0)
P(x,y)
A (r,0)
P(x,y)
A (r,0)
P(x,y)
60or /3 780or 13/3420or 7/3
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COTERMINALANGLES
Two or more angles in standard position arecoterminal angles if the position of P is the
same for each angle.
If represents any angle, then any angle
coterminal with is represented by theseexpressions, where n is an integer.
+ n(360)
+ 2n
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EX1 GIVEN = /6
i) Draw the angle in standard position
ii) Find two other angles which are coterminal with
and illustrate them on the diagrams located on your
next slide.
iii) Write an expression to represent any angle
coterminal with
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A (r,0)
P(x,y)
/6 -11/613/6
EX 1- SOLUTION
A (r,0)
P(x,y)
A (r,0)
P(x,y)
i ii
iii. Any angle coterminal with is represented by the expression /6 + 2n, where n is
an integer
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P is a point on the terminal arm of an angle
in standard position. Explain how you
would determine the quadrant in which P
is located, if you know the value of in:
Degrees
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USE YOUR REASONING
Suppose P has rotated 830 about (0,0) from A.
In which quadrant is P located now?
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P is a point on the terminal arm of an angle
in standard position. Explain how you
would determine the quadrant in which P
is located, if you know the value of in:
Radians
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USE YOUR REASONING
Suppose P has rotated 29 /4 about (0,0) from A.
In which quadrant is P located now?
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WRAPPING FUNCTION
The relation that maps thedistance from A onto the
end point P is formalized in
what we call the wrapping
function.
-- 2
-- 1
-- -1
-- -2
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THE WRAPPING FUNCTION
The wrapping function, W(s), maps any real number, s, onto the
coordinates of a point on the unit circle.
W:s (x,y)
The domain of W(s) is
The range of W(s) is { (x,y) X | x2
+ y2
= 1}.
W(s) is periodic with a period of 2, that is:
W(s) = W(s + 2k), k
W: (x,y)
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B
ecause the measures of trigonometric angles are directed, wealso direct the measures of the arcs.
Positive Angle Measure
Negative Angle Measure
Positive Arc Measure
Negative Arc Measure
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TRIGONOMETRIC POINT P()
P()
y
XO
1
-
-
P(-)
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REFRESHER
P()
y
X
rad
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REFRESHER
X
P()
y
X
1
rad
y
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HOW CAN TAN() BE DEFINED IN THE UNIT
CIRCLE?
P(cos, sin)
y
X
1
rad
y
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HOW CAN TAN() BE DEFINED IN THE UNIT
CIRCLE?
P(cos, sin)
y
X
1
rad
y
Tan = sin = y-coordinate
cos x-coordinate
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USING NEW DEFINITIONS
For an arc of given measure where P() = (cos ,sin ), give the definitions of cosec , sec , and
cot .
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USING NEW DEFINITIONS
For an arc of given measure where P() = (cos ,sin ), give the definitions of cosec , sec , and
cot .
cosec = 1 = 1 sec = 1 = 1
sin y cos x
cot = cos = x
sin y
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COMPLETE THE FOLLOWING CHART
Arc
Measure
Quadrant Sign of
sin
Sign of
cos
Sign of
tan
0 < < /2
/2 < <
< < 3/2
3/2 < < 2
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COMPLETE THE FOLLOWING CHART
Arc
Measure
Quadrant Sign of
sin
Sign of
cos
Sign of
tan
0 < < /2 1 + + +
/2 < < 2 + - -
< < 3/2 3 - - +
3/2 < < 2 4 - + -
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POSITIVE QUADRANTS
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POSITIVE QUADRANTS
II I
III IV
Sin
cosec
ALL
Tan
cot
Cos
sec
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UNIT CIRCLE