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Because the slope of the curve of sin(x) is cos(x). Or, equivalently, the limit of sin(x) over x tends to cos(x) as x tends to zero.
Yes. Except where sin x = 0, because then you would be dividing by zero so the quotient is undefined.
(cos x sin x) / (cos x sin x) = 1. The derivative of a constant, such as 1, is zero.
Zero
There will be 4 possible solutions if all you are looking for is the angles, so you will need to find out which quadrant your angle is in.2sin²x = sin x **Subtract sin x from both sides.2sin²x - sin x = 0 **Then factor out sin x.sin x(2sin x - 1) = 0 **Set each equal to zero. (AB=0 is the same as A=0 OR B=0).sin x = 0 or 2sin x - 1 = 0 to 2sin x = 1 to sin x = 1/2At this point all that is left to do is find out where sin x = 0 or 1/2, which is 0, 180 for sin x = 0 or 30, 150 for sin x = 1/2.