x-y coordinates on a unit circle, using cos/sin

[xxMsJojoxx]

Why is the coordinates for point P (cos α, sin α )? Tringle POB is not a right triangle. -- Why is cosine/sine used to solve for the x and y value of point P? Can you explain how we arrive at (cos α, sin α) as the coordinates?

 

[tkhunny]

The (x,y) coordinates of every point on the Unit Circle are $$(\cos(ANGLE),\sin(ANGLE))$$, where "ANGLE" is the rotation indicated.

In Polar Coordinates, the address of every point on the Unit Circle can be expressed as $$(1,ANGLE)$$.

The address of Q is $$(\cos(\beta),\sin(\beta))\;or\;(1,\beta)$$, in Cartesian and Polar coordinates respectively. No right triangle there, either. Why did you decide there needed to be a right triangle? Very few points could be labeled with that restriction.

 

[Dr.Peterson]

Why is the coordinates for point P (cos α, sin α )? Tringle POB is not a right triangle. -- Why is cosine/sine used to solve for the x and y value of point P? Can you explain how we arrive at (cos α, sin α) as the coordinates?

View attachment 22564

This is simply a definition.

Now, if you started your study of trigonometry with the right-triangle definitions (which apply only to acute angles), then for any acute angle (such as beta in your picture), the right-triangle definitions for sine and cosine imply that the coordinates of point Q are (cos(beta), sin(beta)), right? So it is a natural extension to define the sine and cosine for angles in other quadrants to be the y and x coordinates of the point on the unit circle, as well.

 

[Steven G]

Drop a perpendicular line and there will be a right triangle.