linguistic:voice_change
Unit circle definition

The sine and cosine functions may also be defined in a more general way by using unit circle, a circle of radius one centered at the origin (0, 0) {\displaystyle (0,0)}, formulated as the equation of x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} in the Cartesian coordinate system. Let a line through the origin intersect the unit circle, making an angle of θ {\displaystyle \theta } with the positive half of the x {\displaystyle x} - axis. The x {\displaystyle x} - and y {\displaystyle y} - coordinates of this point of intersection are equal to cos (θ) {\displaystyle \cos(\theta)} and sin (θ) {\displaystyle \sin(\theta)}, respectively; that is, sin (θ) = y, cos (θ) = x. {\displaystyle \sin(\theta)=y,\qquad \cos(\theta)=x.}

This definition is consistent with the right-angled triangle definition of sine and cosine when 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the y {\displaystyle y} - coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}}, even under the new definition using the unit circle.