5 1 The Unit Circle Unit circle the

5. 1 The Unit Circle

�Unit circle – the circle with radius 1 centered at the origin in the xy-plane. The equation is: x 2 + y 2 = 1

EX �Recall: Show that (1, -3) is on the line 2 x + 3 y = -7 EX

Ex

Terminal Points �Start at (1, 0) and move ccw if t is positive and cw if t is negative. �We arrive at the point P(x, y) on the unit circle. P(x, y) is the terminal point determined by the real number t.

�The circumference of the unit circle is C = 2 If a point starts at (1, 0) and moves ccw all the way around and returns to (1, 0), then we have traveled a distance of 2 pi. �Travel half way around = _____ �Travel a quarter of the way around = ______

Ex Find the terminal point on the unit circle determined by each real number t. � Different values of t can determine the same terminal point.

The unit circle is symmetric with respect to the line y = x. Then you can solve a system of equations to find the terminal points. OR you can memorize the table below: �You should’ve already memorized this…

The Reference Number �Let t be a real number. �Similar to a reference angle

EX Find the reference number

Ex Find the terminal points determined by each given real number t.

�Since the circumference is 2 pi, the terminal point determined by t is the same as that determined by t + 2 pi or t – 2 pi. �In general, we can add or subtract 2 pi any number of times without changing the terminal point determined by t. �Coterminal angles have the same terminal point

EX Find the terminal point

Ex Find the terminal points

Ex

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