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Trigonometric Functions of Real Numbers 6.3

Trigonometric Functions of Real Numbers 6.3. The unit O circle. Mrs. Crespo 2011. The Unit Circle. S. S. 1. r. (0,1).

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Trigonometric Functions of Real Numbers 6.3

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  1. Trigonometric Functions of Real Numbers 6.3 The unit O circle Mrs. Crespo 2011

  2. The Unit Circle S S 1 r (0,1) • With radius r=1 and a center at (0,0). • θ= • = • = S r = 1 S = arc length (-1,0) (1,0) (0,0) (0,-1) Mrs. Crespo 2011

  3. The Unit Circle (0,1) • To find the terminal point P(x,y) for a given real number t, move t units on the circle starting at (1,0). P(x,y) t (-1,0) (1,0) (0,0) -t • Move counterclockwise if t > 0. P(x,y) • Move clockwise if t < 0. (0,-1) Mrs. Crespo 2011

  4. The Unit Circle and the Trig. Functions x r y y x 1 y r 1 x x x y x r 1 y r 1 y (0,1) • With radius r=1, then • cott = • sect = • csct = • tant = • sin t = • cost = • = • = • = • = • = y • = x r = 1 y (-1,0) (1,0) (0,0) X (0,-1) Mrs. Crespo 2011

  5. Example 1 y x (0,1) P(-3/5 ,-4/5) is on the terminal side of t. Find sin t, cost, and tan t. • tant = • cost = • sin t = • y • x (-,+) (+,+) (-1,0) (1,0) -4/5 • = (+,-) (-,-) • = -3/5 P(-3/5 ,-4/5) 4/3 • = • = (0,-1) Mrs. Crespo 2011

  6. Your Turn 1 y x (0,1) P(4/5 , 3/5) is on the terminal side of t. Find sin t, cost, and tan t. • tant = • sin t = • cost = • y • x P(4/5 ,3/5) (-1,0) (1,0) 3/5 • = • = 4/5 3/4 • = • = (0,-1) Mrs. Crespo 2011

  7. Example 2 Given the following sketch. (0,1) With P(t) P(t) =(4/5 ,3/5) (-1,0) (1,0) t (0,-1) Mrs. Crespo 2011

  8. Example 2 Given the following sketch. (0,1) Find P(t + π) π = 180˚ 180˚ forms a straight line P(t) =(4/5 ,3/5) On QIII (-,-) t+ π (-1,0) (1,0) • adding π means moving ccw t P(t + π) =(-4/5 ,-3/5) (0,-1) Mrs. Crespo 2011

  9. Example 2 Given the following sketch. (0,1) Find P(t - π) π = 180˚ 180˚ forms a straight line P(t) =(4/5 ,3/5) Still on QIII (-,-) • subtracting π means moving cw (-1,0) (1,0) t-π P(t - π) =(-4/5 ,-3/5) (0,-1) Mrs. Crespo 2011

  10. Example 2 Given the following sketch. (0,1) Find P(-t) -t means moving cw P(t) =(4/5 ,3/5) Reflect on x-axis means x-axis is the mirror line (-1,0) (1,0) t (0,-1) Mrs. Crespo 2011

  11. Mirror Line Samples Mrs. Crespo 2011

  12. Example 2 Given the following sketch. (0,1) Find P(-t) -t means moving cw P(t) =(4/5 ,3/5) Reflect on x-axis means x-axis is the mirror line (-1,0) (1,0) t On QIV (+,-) -t P(-t) =(4/5 ,-3/5) (0,-1) Mrs. Crespo 2011

  13. Example 2 Given the following sketch. (0,1) Find P(-t - π) from -t move cw P(t) =(4/5 ,3/5) P(-t - π) =(-4/5 ,3/5) On QII (-,+) (-1,0) (1,0) t -t -t - π • subtracting π means moving cw (0,-1) π = 180˚ 180˚ forms a straight line Mrs. Crespo 2011

  14. Your Turn 2 Given P(t)=(-8/17 ,15/17) , find: (0,1) a) P(t+ π) b) P(t- π) c) P(-t) P(-t - π)=(8/17 ,15/17) P(t)=(-8/17 ,15/17) d) P(-t- π) (-1,0) (1,0) P(t + π)=(8/17 ,-15/17) P(-t)=(-8/17 ,-15/17) P(t - π)=(8/17 ,-15/17) (0,-1) Mrs. Crespo 2011

  15. The Unit Circle 3π π 2 2 (0,1) We know that: Π = 180˚ 2 Π = 360˚ • 360˚ is one full rotation. π 2π (-1,0) (1,0) (0,0) Then, P(x , y) = P(cost, sin t) (0,-1) Mrs. Crespo 2011

  16. Examples P(x , y) = P(cost, sin t) on the Unit Circle 3π π π 3π 2 2 2 2 Find • cos • sin • sin • cos 0 • = 1 • = -1 • = 0 • = 0 • = -1 • = 1 • = 0 • cos • sin • cos • sin • = 2π π π 2π Mrs. Crespo 2011

  17. The Unit Circle π 5π 2π π 3π π 7π 5π 3π 5π 7π 4π 11π π 3 6 4 3 4 4 3 4 2 3 6 6 6 2 • Degrees (0,1) (1/2 ,√3/2) (-1/2 ,√3/2) • Points 90˚ (√2/2 , √2/2) (-√2/2 , √2/2) Start with QI. • The denominators for all coordinates is 2. • The x-numerators going from 60˚, 45˚ to 30˚, write 1, 2, 3. • The y-numerators going from 30˚, 45˚ to 60˚, write 1,2,3. • Square root all numerators. 120˚ 60˚ 135˚ 45˚ (√3/2 ,1/2) (-√3/2 ,1/2) 150˚ 30˚ π 180˚ 0˚ 0 (-1,0) (1,0) 360˚ 2π 330˚ 210˚ 315˚ (√3/2 ,-1/2) (-√3/2 ,-1/2) 225˚ 300˚ 240˚ • Once QI special angles have points determined, the rests are easy to find out. (√2/2 , -√2/2) (-√2/2 , -√2/2) 270˚ (1/2 ,-√3/2) (-1/2 ,-√3/2) • Radians (0,-1) Mrs. Crespo 2011

  18. Formulas for Negatives sin (-t) = - sin (t) cos (-t) = cos (t) tan (-t) = - tan (t) csc (-t) = - csc (t) sec (-t) = sec (t) cot (-t) = - cot (t) EXAMPLES -√3 -2 2 -1 Mrs. Crespo 2011

  19. Estimating P(x , y) = P(cosθ, sinθ) 0 sin (0) = cos (0)= 1 sin (1) = .02 sin (3) = .05 sin (5) = .09 cos (3) = 1 cos (-6) = 1 cos (4) = 1

  20. Even and Odd Functions Even Functions Odd Functions • The form is f(-x) = f(x). • Signs of both coordinate points change. • Symmetric with respect to y-axis. • The form is f(-x) = - f(x). • Signs of y-coordinates do not change. • Symmetric with respect to the origin. sin (-t) = - sin (t) cos (-t) = cos (t) tan (-t) = - tan (t) csc (-t) = - csc (t) sec (-t) = sec (t) cot (-t) = - cot (t) • TURN TO PAGE 441 AND OBSERVE THE GRAPHS ON THE TABLE. Mrs. Crespo 2011

  21. Homework • PAGE 444 : 1- 20 ODD Mrs. Crespo 2011

  22. Resources • Textbook: Algebra and Trigonometry with Analytic Geometry by Swokowski and Cole (12th Edition, Thomson Learning, 2008). • http://www.mathlearning.net/learningtools/Flash/unitCircle/unitCircle.html • http://www.mathvids.com/lesson/mathhelp/36-unit-circle • www.embeddedmath.com/downloads • tutor-usa.com/video/lesson/trigonometry/​4059-unit-circle. • PowerPoint and Lesson Plan customization by Mrs. Crespo 2011. • Ladywood High School Mrs. Crespo 2011

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