In an attempt to show that a unitary complex $z$ must be of the form $z=\cos \alpha + i \sin \alpha $ for some $\alpha \in \mathbb R$ we are led to this situation: Let $x$ and $y$ be reals such that $x^2+y^2=1$, why there must be some $\alpha\in \mathbb R$ such that $x=\cos \alpha$ and $y=\sin \alpha$ ? Thank you for your help !
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$\begingroup$ Is it acceptable to you that, except for $x=0$, there is always some value of $\alpha$ such that $\tan\alpha = y/x$? $\endgroup$– peterwhySep 4, 2015 at 22:05
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On a rectangular coordinates, draw a unit circle $x^2 + y^2 = 1$. From the given condition, a point $P(x_0, y_0)$ satisfies $x_0^2+y_0^2 = 1$, and the point lies on the circle.
Join the origin $O$ and $P$, and measure the angle anticlockwise from the positive $x$-axis to line $OP$. Call the angle $\alpha$. By definition of $\sin$ and $\cos$, the coordinates of $P$ can be written as $(\cos\alpha, \sin\alpha)$.
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$\begingroup$ How is that by definitin of $\sin$ and $\cos$? $\endgroup$– palioSep 4, 2015 at 22:16
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$\begingroup$ @palio Wikipedia: In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system. Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The point's distance from the origin is always 1. $\endgroup$– peterwhySep 4, 2015 at 22:20