That is the definition. If you take your unit circle (a circle with radius 1 centered at the origin (0,0). you start at (1,0) and go counterclockwise around the circle 90° you end up at (0,1) that 0 is the cosine of the angle 90°
In fact, you don't even need the unit circle. Take a circle of any radius r, and draw a ray at 90 degrees. This will intersect the y-axis. So as above, the coordinates are (0,r) (instead of (0,1)) so cos(90 degrees)=x/r=0/r=0
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Cosine (0) = 1 Sin(0) = 0 The sine and cosine curves are two intertwining curves, that complement each other, hence the words 'Sine (Curve) and Cosine ( COmplementary Curve).
The Y-Intercept of the cosine function is X = 0, Y = cosine(0) = 1.
sin(30) = sin(90 - 60) = sin(90)*cos(60) - cos(90)*sin(60) = 1*cos(60) - 0*sin(60) = cos(60).
Cos(0) = 1
The angles in quadrant one measure between 0 degrees and 90 degrees. In radians, that's between 0 and pi/2. Quadrant one is the quadrant where both X and Y (or cosine theta and sine theta) are positive.