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4C.FTF.3.4.12.8.11 2011 Domain: Functions Cluster: Extend the domain of trigonometric functions using the unit circle. Standards: 3.(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number. 4.(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Essential Questions Enduring Understandings Why do we create the unit circle using special right triangles? Why do we use the unit circle? How can symmetry be used to explain mathematical concepts? Content Statements Students will solve problems involving the trigonometric functions of real numbers and the properties of the sine and cosine as periodic functions. Students will generate the graphs of sine and cosine functions. Students will use the unit circle to solve trigonometric functions. Mathematical equations can be solved using inverse trigonometric functions. There is a relationship between sine and cosine and the unit circle. Symmetry and periodicity can be explained using the unit circle. Activities, Investigation, and Student Experiences Use the unit circle to find each value. 1. tan 180° 2. sec (-90°) 3. csc 30° 4. cot (-180°) 5. sin 225° 6. cos 300° 7. tan π/2 8. cot 5π/4 9. sin 7π/4 10. csc 7π/6 Find the amplitude, period, and phase shift of each function. Graph each function, showing at least one period. a. y = 4sin(2x – π) b. y =2cos(2πx -4) 4C.FTF.3.4.12.8.11 Students will differentiate between odd and even functions. Assessments Example problems: Find the period of each function. y = -2 sin x y = 4 cos (x/3) y = 1.5 cos 4x y = -2/3 sin (x/2) Example open-ended problem. The function P = 100 – 20cos(5πt)/3 approximates the blood pressure P in millimeters of mercury at time t in seconds for a person at rest. a) Find the period of the function. b) Find the number of heart beats per minute. Example graphing problem. Sketch the graph of the function (include two full periods). a) y = -2 sin 6x b) y = -3 cos 4x c. y = -3sin(2x + π/2) 2011 4C.FTF.3.4.12.8.11 Equipment Needed: Calculator (graphing) Smart board White board Overhead 2011 Teacher Resources: http://www.mathsisfun.com/geometry/unit-circle.html http://mac.iupui.edu/unitcircle154.pdf http://www.youtube.com/watch?v=ao4EJzNWmK8 http://mathworld.wolfram.com/Trigonometry.html http://www.nabla.hr/Z_TrigonometricFunctions-A.htm http://www.analyzemath.com/trigonometry/periods_trig onometric.html http://www.analyzemath.com/trigonometry/periods_trig onometric.html http://www.teacherschoice.com.au/maths_library/functio ns/about_trigonometric_functions.htm http://www.onlinemathlearning.com/graphing-trigfunctions.html http://www.sosmath.com/CBB/viewtopic.php?t=596 http://www.analyzemath.com/Trigonometry.html