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Chapter 5 – Trigonometric Functions: Unit Circle Approach 5.1 - The Unit Circle The Unit Circle The unit circle is the circle of radius 1 centered at the origin in the xy-plane. Its equation is x y 1 2 2 5.1 - The Unit Circle Example – pg. 375 Show that the point is on the unit circle. 5 12 4. , 13 13 5 2 6 6. , 7 7 11 5 8. , 6 6 5.1 - The Unit Circle Terminal Points on the Unit Circle Suppose t is a real number. Let t be the distance along the unit circle starting at the point (1, 0) and ending at the point P (x, y). This point is the terminal point determined by the real number t. 5.1 - The Unit Circle Terminal Points The circumference of a circle is C = 2. So if a point starts at (1, 0) and moves counterclockwise all the way around the circle, it travels a distance of t = 2. 5.1 - The Unit Circle Terminal Points The circumference of a circle is C = 2. So if a point starts at (1, 0) and moves counterclockwise half of the way around the circle, it travels a distance of 1 t 2 2 5.1 - The Unit Circle Terminal Points The circumference of a circle is C = 2. So if a point starts at (1, 0) and moves counterclockwise a quarter of the way around the circle, it travels a distance of 1 t 2 4 2 5.1 - The Unit Circle Terminal Points The circumference of a circle is C = 2. So if a point starts at (1, 0) and moves counterclockwise three quarters of the way around the circle, it travels a distance of 3 3 t 2 4 2 5.1 - The Unit Circle Terminal Points 5.1 - The Unit Circle Examples – pg. 376 Find the terminal point P (x, y) on the unit circle determined by the given value of t. 3 24. t 2 7 26. t 6 27. t 3 5.1 - The Unit Circle Example – pg. 376 Suppose that the terminal point determined by t is the point on the unit circle. Find the terminal point determined by each of the following. (a) -t (b) - t (c) 4 + t (d) t - 5.1 - The Unit Circle Reference Number Let t be a real number. The reference number`t associated with t is the shortest distance along the unit circle between the terminal point determined by t and the x-axis. 5.1 - The Unit Circle Examples – pg. 376 Find the reference number for each value of t. 5.1 - The Unit Circle Using Reference Numbers to Find Terminal Points To find the terminal point P by any value of t, we use the following steps: 1. Find the reference number`t. 2. Find the terminal point Q (a, b) determined by`t. 3. The terminal point determined by t is P (±a, ±b), where the signs are chosen according to the quadrant in which this terminal point lies. 5.1 - The Unit Circle Examples – pg. 376 Find (a) the reference number for each value of t and (b) the terminal point determined by t. 7 42. t 3 7 44. t 6 13 46. t 6 17 48. t 4 31 50. t 6 41 52. t 4 5.1 - The Unit Circle