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Understanding the Unit Circle and Trigonometric Functions

This resource explores the unit circle, its properties, and the connection to trigonometric functions. Specifically, it details how to find the terminal point (P(x,y)) for a given angle (t) by moving around the circle. With a radius of 1 and centered at the origin, it explains how to compute sine, cosine, and tangent for various points on the unit circle. It also includes examples for finding values of trigonometric functions at specific angles and their behavior under transformations. Ideal for students seeking to grasp fundamental concepts of trigonometry.

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Understanding the Unit Circle and Trigonometric Functions

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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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