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bmed90
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As the title inquires, I am curious as to how or why the Sin function represents y coordinate on the unit circle.
bmed90 said:I guess what I am asking is why is sin defined as the y coordinate?
The y coordinate is interesting in some mathematical calculations, so this function got its own name ("sine"). Other parameters got other names (like cosine, tangent, ...).bmed90 said:I guess what I am asking is why is sin defined as the y coordinate?
phinds said:Do you understand what a "sin" function is? Can you DRAW a unit circle and see what you would get if you took the sin of some arbitrary radius line?
willem2 said:Consider the triangle formed by the origin, a point on the unit circle in the first quadrant at an angle A, and the projection of this point on the x-axis. This is a right-angled triangle, with hypothenuse 1. since sin A = (opposite side of A)/(hypothenuse), the opposite side is sin A. This is also the y-coordinate of the point on the unit circle.
Reflection around the x- and y-axis will prove this for the other quadrants as well.
Sin, or the sine function, is used to represent the vertical component (Y-coordinate) of a point on the unit circle because it is defined as the ratio of the opposite side of a right triangle to its hypotenuse. On the unit circle, the hypotenuse is always equal to 1, making it a perfect function to represent the Y-coordinate.
To calculate Sin on the unit circle, you need to first determine the angle measure in radians. Then, you can use the formula Sin(θ) = opposite/hypotenuse, where θ is the angle measure. This will give you the Y-coordinate of the point on the unit circle.
The unit circle is a powerful tool in mathematics because it allows us to easily visualize and understand the relationship between the trigonometric functions (such as Sin) and the coordinates of a point on the circle. It also helps us to understand the periodic nature of these functions.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. On the unit circle, the hypotenuse is always equal to 1, so the Pythagorean theorem simplifies to c^2 = a^2 + b^2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides. This relationship is fundamental in understanding the trigonometric functions on the unit circle.
Yes, Sin can be used to represent the Y-coordinate of a point on any circle, not just the unit circle. It can also be used to represent the vertical component of a point on a graph or any other shape that involves right triangles. However, the unit circle is a specific case where the relationship between Sin and the Y-coordinate is simplified and easily understood.