I'd like to see the flaw in my logic in the following:
I have a circle with radius 1.
Therefore:
opposite side = sin(angle) = opposite / hypotenuse = opposite / 1
See this picture for a graphic depiction.
Therefore, the opposite sides (in green on the picture) when changing the x value from 1 to 0 will increase in height and their co-ordinates effectively mimic the circle's curve.
Because the height of these opposite sides equals the sine of the angles, these can be mapped onto a sine graph (x-axis is the angles in degrees, y-axis is opposite side height), and should replicate the circle's curve but mirrored. This means that sine graphs should have a semicircle shape above the x-axis from x values of 0-180 and a mirrored semicircle below the x-axis from x values of 180-360.
Where have I gone wrong?
When I look at a real sine graph I can't cut out the bottom section, slide it under the positive parabola and form a circle – but why not?
Best Answer
It's not standard to answer a question with an image, but I think the image says more than 1000 words in this case:
The point is that what you are drawing on the x axis is the angle, not the length of one of the sides of the triangle. The angle is proportional to the length of the circle section.
Image Source. Credit for the image goes to Lucas V. Barbosa.