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THE UNIT CIRCLE

THE UNIT CIRCLE. Lesson 4.2. A circle with center at (0, 0) and radius 1 is called a unit circle. . The equation of this circle would be . (0,1). (-1,0). (1,0). (0,-1).

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THE UNIT CIRCLE

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  1. THE UNIT CIRCLE Lesson 4.2

  2. A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be (0,1) (-1,0) (1,0) (0,-1)

  3. Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value? x = 1/2 (0,1) (-1,0) (1,0) (0,-1)

  4. Now, draw a right triangle from the origin to the point when x = ½. Use this triangle to find the values of sin, cos, and tan. (0,1) (-1,0) (1,0)  (0,-1)

  5. For any point on the unit circle: Sine is the y value Cosine is the xvalue. Tangent is

  6. Find the sin, cos, and tan of the angle (0,1) (-1,0) (1,0)  (0,-1)

  7. How many degrees are in each division? 90° 135° 45° 180° 45° 0° 225° 315° 270°

  8. How about radians? 90° 135° 45° 180° 0° 225° 315° 270°

  9. How many degrees are in each division? 90° 120° 60° 150° 30° 180° 30° 0° 210° 330° 240° 300° 270°

  10. How about radians? 90° 120° 60° 150° 30° 180° 30° 0° 210° 330° 240° 300° 270°

  11. Let’s think about the function f() = sin  All real numbers. Domain: -1  sin   1 Range: (0, 1) (1, 0) (-1, 0) (0, -1)

  12. Let’s think about the function f() = cos  All real numbers Domain: -1  cos   1 Range: (0, 1) (-1, 0) (1, 0) (0, -1)

  13. Let’s think about the function f() = tan  x ≠ n( /2), where n is an odd integer Domain: Range: All real numbers

  14. Trig functions:

  15. Look at the unit circle and determine sin 420°. Sine is periodic with a period of 360° or 2. So sin 420° = sin 60°.

  16. This would have the same value as Reciprocal functions have the same period. PERIODIC PROPERTIES sin( + 2) = sin cosec( + 2) = cosec  cos( + 2) = cos  sec( + 2) = sec  tan( + ) = tan  cot( + ) = cot  1 Subtract the period until you reach an angle measure you know.

  17. Positive vs. Negative & Even vs. Odd Even if: f(-x) = f(x) Odd if: f(-x) = -f(x) Remember negative angle means to go clockwise

  18. Cosine is an even function.

  19. Sine is an odd function.

  20. EVEN-ODD PROPERTIES sin(-  ) = - sin (odd)cosec(-  ) = - cosec (odd) cos(-  ) = cos  (even) sec(-  ) = sec (even) tan(-  ) = - tan (odd) cot(-  ) = - cot (odd) Problem Set 4.2

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