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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


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