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..." ???
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.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. "
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!
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).
"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?
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.
"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?