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Math Help - Why is sine negative

  1. #1
    Member wiseguy's Avatar
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    Why is sine negative

    Here is what I wrote:


    "Sine is an odd function because whenever it passes under the x-axis, it becomes negative.

    A 3-4-5 (point (3,4) on a circle) triangle in the first quadrant would have a sine value of sin(4/5). If the triangle were reflected over the x-axis, the sine value would change to sin(-4/5).

    Another rationale why sine is negative is because it has no symmetry over the x or y axis. Cosine is symmetric over the y-axis, thus it is even. "


    I'm not sure if this is answering the question, or if it is too "far out..." ???
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by wiseguy View Post
    Here is what I wrote:


    "Sine is an odd function because whenever it passes under the x-axis, it becomes negative.

    A 3-4-5 (point (3,4) on a circle) triangle in the first quadrant would have a sine value of sin(4/5). If the triangle were reflected over the x-axis, the sine value would change to sin(-4/5).

    Another rationale why sine is negative is because it has no symmetry over the x or y axis. Cosine is symmetric over the y-axis, thus it is even. "


    I'm not sure if this is answering the question, or if it is too "far out..." ???
    Why is the sine of what negative?

    (what you have written seems largle waffle and gobbledy-gook)

    CB
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  3. #3
    Member Mathelogician's Avatar
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    Lightbulb

    Here is what I wrote:
    "Sine is an odd function because whenever it passes under the x-axis, it becomes negative.

    A 3-4-5 (point (3,4) on a circle) triangle in the first quadrant would have a sine value of sin(4/5). If the triangle were reflected over the x-axis, the sine value would change to sin(-4/5).

    Another rationale why sine is negative is because it has no symmetry over the x or y axis. Cosine is symmetric over the y-axis, thus it is even. "
    No, this point show an angle x which its sine is 4/5; that is sin(x)=4/5 and by the reflection about x-axis x becomes -x and then you have sin(-x)=-4/5.
    Sine is symmetric with respect to the point O(0,0) [which is the origin of the cartesian plane]; Therefore it's odd. And for cosine you are right.
    Note that a function is odd if f(-x)=-f(x); Therefore not all functions which have no symmetry over the x or y axis are odd!
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  4. #4
    Member wiseguy's Avatar
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    Sorry, meant to say why is sine odd
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  5. #5
    Member wiseguy's Avatar
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    I edited my response... how does it look?



    Sine is an odd function because whenever it passes under the
    x-axis, it becomes negative.

    A 3-4-5 (point (3,4) on a circle) triangle in the first
    quadrant would have a sine value of sin(x)=(4/5). If the
    triangle
    were reflected over the x-axis, the sine value would change to
    sin(x)=(-4/5).
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  6. #6
    Member Mathelogician's Avatar
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    I think it would be better to not use a triangle for your sentence. Because a negative sine is not usual for a triangle!
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  7. #7
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    Quote Originally Posted by wiseguy View Post
    I edited my response... how does it look?



    Sine is an odd function because whenever it passes under the
    x-axis, it becomes negative.

    A 3-4-5 (point (3,4) on a circle) triangle in the first
    quadrant would have a sine value of sin(x)=(4/5). If the
    triangle
    were reflected over the x-axis, the sine value would change to
    sin(x)=(-4/5).
    An odd function has the point of intersection of the x and y axes (the origin) as a centre of symmetry.
    This is true of sinx.

    An even function has the y axis as an axis of symmetry.
    This is true of cosx.

    You may think in terms of a right-angled triangle placed within the circle.

    Another way to consider it is to use a unit-radius circle centred at the origin.

    Then sin(angle) gives the vertical co-ordinate of a point on the circumference.
    A positive angle is anti-clockwise and a negative angle is clockwise.

    Hence, the vertical co-ordinate for an anticlockwise angle A has opposite sign to
    the vertical co-ordinate for an angle of the same magnitude in the clockwise direction.

    Hence sin(A)=-sin(-A).
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  8. #8
    Member wiseguy's Avatar
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    "Sine is an odd function because when A in sin(A)=y becomes negative, y becomes negative. Thus, whenever sine passes below the x axis, it is always negative.

    In a unit circle, the y-coordinate of a point is always denoted by sinθ. Whenever sine passes below the x axis, it is always negative."


    I think this might cover it?
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  9. #9
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    Quote Originally Posted by wiseguy View Post
    "Sine is an odd function because when A in sin(A)=y becomes negative, y becomes negative.

    ********Thus, whenever sine passes below the x axis, it is always negative******

    In a unit circle, the y-coordinate of a point is always denoted by sinθ.

    ********Whenever sine passes below the x axis, it is always negative."**********


    I think this might cover it?
    Those 2 lines between asterisks need work.

    You need to come to the point where you can explain why Sin(A)=-Sin(-A).
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  10. #10
    Member wiseguy's Avatar
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    Okay, Sin(A)=-Sin(-A) makes perfect sense because as A crosses below the x axis, sin(-a)=negative. So, sin(a)=-sin(-a), because sin(-a) is negative, adding another negative sign makes it positive...
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  11. #11
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    Quote Originally Posted by wiseguy View Post
    Okay, Sin(A)=-Sin(-A) makes perfect sense because as A crosses below the x axis, sin(-a)=negative. So, sin(a)=-sin(-a), because sin(-a) is negative, adding another negative sign makes it positive...
    Yes, that's getting better.

    Using the definitions of Sine, Cosine and Tangent in the right-angled triangle,
    side lengths can be calculated from angles and vice versa, so all dimensions are positive.

    In the circle, the triangle can be placed on the x (or y) axis to calculate distances,
    however in the circle we are locating position, hence the introduction of negative values
    to denote position relative to the origin (left or right...up or down).

    Hence if the angle is between 0 and 180 degrees, sin(angle) is +
    since it points out the vertical co-ordinate of a point on the circumference
    (just think of the value of the co-ordinate as a point on the vertical axis itself).

    Then if we make the exact same angle but in a clockwise (negative) direction,
    the angle will be denoted a negative angle
    and the vertical co-ordinate will be the negative of the previous one.
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  12. #12
    Member wiseguy's Avatar
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    "Sine is an odd function because when A in sin(A)=y becomes
    negative, y becomes negative.

    In a unit circle, the y-coordinate of a point is always
    denoted by sinθ.

    Sin(A) equals -sin(-A) because as A crosses below the x axis, sin(-A) equals a negative value. Thus, adding a negative sign to sin(-A) creates a positive, rendering it equal to sin(A)."

    I'm going to fly with this. How does it look?
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  13. #13
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    Quote Originally Posted by wiseguy View Post
    "Sine is an odd function because when A in sin(A)=y becomes
    negative, y becomes negative.

    *******Not only is it negative, it's the negative of sin(A).
    You need to understand that if sin(A)=k, then sin(-A)=-k.

    For example... sin\left(30^o\right)=0.5\ and\ sin\left(-30^o\right)=-0.5***************

    In a unit circle, the y-coordinate of a point is always
    denoted by sinθ. *****the unit-circle is centred at the origin (0,0)*****

    Sin(A) equals -sin(-A) because as A crosses below the x axis, sin(-A) equals a negative value.

    *****the exact negative of sin(A), if ... 0<A<180^o********

    Thus, adding a negative sign to sin(-A) creates a positive, rendering it equal to sin(A)."

    ****the last line is not so important because you can also have angles > 180 degrees*******

    I'm going to fly with this. How does it look?
    If 180<A<360 degrees, sin(A) is negative.
    sin(-A) will be the negative of that, ie positive.
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