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Why value of sine and cosine theta remain between 1 and -1 from angle 0° to 360°?? Is it any specific reason for it... i need answer

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closed as off-topic by Henrik, Parcly Taxel, Arnaldo, Davide Giraudo, Yujie Zha May 12 '17 at 13:44

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    $\begingroup$ The $\sin, \cos$ of an angle are the ratios of the legs of a right triangle to the hypotenuse. The legs can not be longer than the hypotenuse. $\endgroup$ – lulu May 12 '17 at 11:24
  • $\begingroup$ I would say that lulu's answer pretty much sums it up. $\endgroup$ – Matti P. May 12 '17 at 11:27
  • $\begingroup$ I cannot understand your answer .. please explain it $\endgroup$ – WaQar ALi May 12 '17 at 11:29
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The simple reason is that the length of the sides of a right triangle are always less than the length of the hypotenuse. So, the ratio of any side and hypotenuse is always less than 1. Now, in the case of 0 or 90 degree, the figure is no longer a triangle because one of the sides ceases to exist while the other side coincides with the hypotenuse. This is the reason that in these extreme cases, sin and cosine take values 0 or 1.

This explanation doesn't take into account the negative values. On a higher level, cosine and sine of an angle are defined respectively as the x and y co-ordinates of the point that you reach on a unit circle (with origin as center) by rotating through that angle. Since the circle is a unit circle with center at the origin, the co-ordinates on that circle can never be greater than 1 or less than -1.

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In case of right triangle, cos of an angle is ratio of side adjacent to that angle to hypotenuse. So, cos of an angle is basically,a ratio. Numerator of this ratio is clearly smaller than denominator as hypotenuse is bigger than any other side in right triangle. So a proper fraction is always less than 1. In case of 2nd, 3rd quadrants it is greater than -1. At quadrant angles it is -1 or 1. So cos of anvl angle lies between 1 and -1

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