File:Circle cos sin.gif
Circle_cos_sin.gif (650 × 390 pixels, file size: 517 KB, MIME type: image/gif, looped, 60 frames, 6.0 s)
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Summary
DescriptionCircle cos sin.gif |
English: We have the unit circle (with radius = 1) in green, placed at the origin at the bottom right.
In the middle of this circle, in yellow, is represented the angle theta (θ). This angle is the amount of counter-clockwise rotation around the circle starting from the right, on the x-axis, as illustrated. An exact copy of this little angle is shown at the top right, as a visual illustration of the definition of θ. At this angle, and starting at the origin, a (faint) green line is traced outwards, radially. This line intersects the unit circle at a single point, which is the green point spinning around at a constant rate as the angle θ changes, also at a constant rate. The vertical position of this point is projected straight (along the faint red line) onto the graph on the left of the circle. This results in the red point. The y-coordinate of this red point (the same as the y-coordinate of the green point) is the value of the sine function evaluated at the angle θ, that is:
As the angle θ changes, the red point moves up and down, tracing the red graph. This is the graph for the sine function. The faint vertical lines seen passing to the left are marking every quadrant along the circle, that is, at every angle of 90° or π/2 radians. Notice how the sine curve goes from 1, to zero, to -1, then back to zero, at exactly these lines. This is reflecting the fact sin(0) = 0, sin(π/2) =1, sin(π) = 0 and sin(3π/ 2) -1 A similar process is done with the x-coordinate of the green point. However, since the x-coordinate is tilted from the usual convention to plot graphs (where y = f(x), with y vertical and x horizontal), an “untilt” operation was performed in order to repeat the process again in the same orientation, instead of vertically. This was represented by a “bend”, seen on the top right. Again, the green point is projected upwards (along the faint blue line) and this “bent” projection ends up in the top graph’s rightmost edge, at the blue point. The y-coordinate of this blue point (which, due to the “bend” in the projection, is the same as the x-coordinate of the green point) is the value of the cosine function evaluated at the angle θ, that is:
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Date | ||||
Source | Own work | |||
Author | Lucas Vieira | |||
Permission (Reusing this file) |
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Captions
some value
16 March 2014
image/gif
c06fd6a5f798c5ac8eb37784d218a2f190916f4d
529,197 byte
5.999999999999995 second
390 pixel
650 pixel
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 09:32, 24 June 2014 | ![]() | 650 × 390 (517 KB) | wikimediacommons>LucasVB | Reverted to version as of 15:59, 16 March 2014 |