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Trigonometric Functions of Real Numbers 6.3 Mrs. Crespo 2011

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

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(0,0) (0,1) (1,0)(-1,0) (0,-1) The Unit Circle 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) Move counterclockwise if t > 0. Move clockwise if t < 0. -t t P(x,y) Mrs. Crespo 2011

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The Unit Circle and the Trig. Functions With radius r=1, then x r cos t = x 1 = = x y r sin t = y 1 = = y y x tan t = r y csc t = 1 y = r x sec t = 1 x = x y cot t = Mrs. Crespo 2011 r = 1 (0,0) (0,1) (1,0)(-1,0) (0,-1) y X

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Example 1 P ( -3 / 5, -4 / 5 ) is on the terminal side of t. Find sin t, cos t, and tan t. Mrs. Crespo 2011 (0,1) (1,0)(-1,0) (0,-1) (+,+) (+,-) (-,-) (-,+) P( -3 / 5, -4 / 5 ) sin t = y= -4 / 5 -3 / 5 cos t = x= y x tan t = == 4/34/3

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Your Turn 1 P ( 4 / 5, 3 / 5 ) is on the terminal side of t. Find sin t, cos t, and tan t. Mrs. Crespo 2011 (0,1) (1,0)(-1,0) (0,-1) P( 4 / 5, 3 / 5 ) sin t = y= 3/53/5 4/54/5 cos t = x= y x tan t = == 3/43/4

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

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

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

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

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Mirror Line Samples Mrs. Crespo 2011

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

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

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

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The Unit Circle We know that: Π = 180˚ (0,0) (0,1) (1,0)(-1,0) (0,-1) 2 Π = 360˚ 360˚ is one full rotation. 2π2π π π 2 3π 2 Mrs. Crespo 2011 Then, P(x, y) = P(cos t, sin t)

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Examples Find Mrs. Crespo 2011 = cos π 2 sin π 2 3π 2 P(x, y) = P(cos t, sin t) on the Unit Circle cos 3π 2 cos π sin π 2π cos 2π2π = = = = = = =

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The Unit Circle (0,1) (1,0)(-1,0) (0,-1) 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. 2π2π π 2 Mrs. Crespo ˚ 30˚ 60˚ 0˚180˚ 90˚ 120˚ 150˚ 360˚ 330˚ 300˚ 210˚ 240˚ 270˚ 135˚ 315˚ 225˚ π 0 2π2π 3 π 4 7π7π 4 5π5π 3 3π3π 2 π 3 π 6 5π 6 3π 4 5π 4 7π 6 4π 3 11π 6 (- 1 / 2, √ 3 / 2 ) ( √2 / 2, √2 / 2 ) ( √3 / 2, 1 / 2 ) (- √3 / 2,- 1 / 2 ) ( √3 / 2,- 1 / 2 ) ( √2 / 2, -√2 / 2 ) ( 1 / 2,- √ 3 / 2 ) ( 1 / 2, √ 3 / 2 ) (- √2 / 2, √2 / 2 ) (- √3 / 2, 1 / 2 ) (- √2 / 2, -√2 / 2 ) (- 1 / 2,- √ 3 / 2 ) Once QI special angles have points determined, the rests are easy to find out. Degrees Points Radians

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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) Mrs. Crespo 2011 EXAMPLES √3

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Estimating sin (0) = P(x, y) = P(cos θ, sin θ ) sin (1) = cos (3) = cos (-6) = cos (4) = sin (5) = cos (0)= 0 sin (3) =

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Even and Odd Functions Even FunctionsOdd 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. Mrs. Crespo 2011 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.

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Homework Mrs. Crespo 2011 PAGE 444 : ODD

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Resources Textbook: Algebra and Trigonometry with Analytic Geometry by Swokowski and Cole (12 th Edition, Thomson Learning, 2008). tCircle.html tCircle.html tutor-usa.com/video/lesson/trigonometry/4059-unit-circle. PowerPoint and Lesson Plan customization by Mrs. Crespo Ladywood High School Mrs. Crespo 2011

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