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Section 5.1 The Unit Circle

Chapter 5 – Trigonometric Functions: Unit Circle Approach. Section 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. Example – pg. 375. Show that the point is on the unit circle.

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Section 5.1 The Unit Circle

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  1. Chapter 5– Trigonometric Functions: Unit Circle Approach Section 5.1 The Unit Circle 5.1 - The Unit Circle

  2. The Unit Circle • The unit circle is the circle of radius 1 centered at the origin in the xy-plane. Its equation is 5.1 - The Unit Circle

  3. Example – pg. 375 • Show that the point is on the unit circle. 5.1 - The Unit Circle

  4. 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

  5. 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

  6. 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 5.1 - The Unit Circle

  7. 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 5.1 - The Unit Circle

  8. 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 5.1 - The Unit Circle

  9. Terminal Points 5.1 - The Unit Circle

  10. Examples – pg. 376 • Find the terminal point P (x, y) on the unit circle determined by the given value of t. 5.1 - The Unit Circle

  11. 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 (c) 4 + t (b)  - t (d) t -  5.1 - The Unit Circle

  12. Reference Number • Let t be a real number. The reference number`tassociated 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

  13. Examples – pg. 376 • Find the reference number for each value of t. 5.1 - The Unit Circle

  14. Using Reference Numbers to Find Terminal Points • To find the terminal point P by any value of t, we use the following steps: • Find the reference number`t. • Find the terminal point Q (a, b) determined by`t. • 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

  15. Examples – pg. 376 • Find (a) the reference number for each value of t and (b) the terminal point determined by t. 5.1 - The Unit Circle

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