Trigonometric Functions of Real Numbers 6 3 THE

Trigonometric Functions of Real Numbers 6. 3 THE UNIT O CIRCLE Mrs. Crespo 2011

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

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

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

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

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

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

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

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

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

Mirror Line Samples Mrs. Crespo 2011

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

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

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

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

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

The Unit Circle Degrees Points 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. Once QI special angles have points determined, the rests are easy to find out. (0, 1) (-1/2 , √ 3/2) (1/2 , √ 3/2) 2π 90˚ π π √ 2 2 2 √ 2 3π 3120˚ 2 3 π 2 2 60˚ 4 4 45˚ 5π 135˚ π √ 3 1 2 2 √ 3 1 6 6 2 2 150˚ 30˚ ( / , /) (- / , / ) (-1, 0) π 180˚ (-√ 3/2 , -1/2) 7π 210˚ 6 225˚ 5π 240˚ 4 4π (-√ 2/2 , -√ 2/2) 3 (-1/2 , -√ 3/2) 0˚ 0 (1, 0) 360˚ 2π 330˚ 11π 6 315˚ √ 3 1 7π 2 2 300˚ 4 5π √ 2 -√ 2 3 2 2 ( / , - / ) 3π 2 270˚ ( / , (1/2 , -√ 3/2) /) (0, -1) Radians Mrs. Crespo 2011

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

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

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

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

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