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Here is a circle that is centered at and that has a radius of 1 unit.
What are the exact coordinates of if is rotated counterclockwise radians from the point ? Explain or show your reasoning.
What are the exact coordinates of point if it is rotated radians counterclockwise from the point ? Explain or show your reasoning.
A circle with center O at the origin of an x y plane. Point P lies on the outside of the curve, in the first quadrant, and is closer to the y axis than the x axis. Point Q lies on the outside of the circle, in the second quadrant, and is closer to the y axis than the x axis. The angle from the x axis to the right of the origin to the segment O Q is labeled with an arc.
Let’s say we have a point with coordinates on the unit circle, like the one shown here:
Using the Pythagorean Theorem, we know that . We also know this is true using the equation for a circle with radius 1 unit, , which is true for the point since it is on the circle.
Another way to write the coordinates of is to use the angle , which gives the location of on the unit circle relative to the point . Thinking of this way, its coordinates are . Since and , we can return to the Pythagorean Theorem and say that is also true.
What if were a different angle and wasn’t in Quadrant I? It turns out that no matter the quadrant, the coordinates of any point on the unit circle given by an angle are . In fact, the definitions of and are the - and -coordinates of the point on the unit circle radians counterclockwise from . Up until today, we’ve only been using the Quadrant I values for cosine and sine to find side lengths of right triangles, which are always positive.
This revised definition of cosine and sine means that is true for all values of defined on the unit circle and is known as the Pythagorean Identity.
The identity relating the sine and cosine of a number. It is called the Pythagorean identity because it follows from the Pythagorean theorem.