5.1 The Unit Circle

Unit circle – the circle with radius 1 centered at the origin in the xy-plane. The equation is:

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

You should’ve already memorized this… 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 Similar to a reference angle 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 + 2pi or t – 2pi. In general, we can add or subtract 2pi 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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