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GBK Precalculus Jordan Johnson

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Today’s agenda Greetings Review / Submit HW: ▫From Section 2-2: Q1-Q10 Problems 1, 5, 9, 19, 21, 25, 27, 29, 30. Warm-up: Transforming sine waves Lesson: Defining Sine & Cosine Classwork / Homework Clean-up

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Right Triangle Trigonometry Prior definitions of sine, cosine, and tangent come from right triangles: ▫sin = a/c ▫cos = b/c ▫tan = a/b a b c

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Extending Trig Definitions Given a right triangle on the Cartesian plane with one acute angle in standard position: ▫Draw a circle of radius 1 centered at the origin. (Called a “unit circle.”) ▫Let (u,v) be the point where the circle intersects the terminal side. (u,v) 1 O

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Extending Trig Definitions Given a right triangle on the Cartesian plane with one acute angle in standard position: ▫Opposite α is v ▫Adjacent to α is u (u,v) 1 O

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Extending Trig Definitions We can modify our definitions accordingly: ▫Given an angle θ in standard position, whose terminal side intersects a circle of radius r at the point (u,v), we define: sin θ = v/r cos θ = u/r Now we have definitions that work for θ >= 90° and θ <= 0°!

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Extending Trig Definitions Our new definitions work because sin(θ) comes from the sine of θ’s reference angle. With this θ, we have: (u,v) r O

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Questions – write answers in notes As θ increases… ▫When is sin θ increasing? Why? ▫When is it decreasing? Why? For what θ-values is sin θ positive / negative? Why? At what value of θ does the sine function start repeating? Explain why. Explain the relationship between the central angle (θ in the unit circle) and the x-coordinate when sine or cosine is plotted as a wave in the xy-plane.

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Summary New definitions of sine & cosine sine & cosine for angles outside (0,90) ▫Based on the unit circle ▫Textbook refers to the unit circle as being in the “uv-coordinate system”.

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Periodicity Recall: ▫At what value of θ does the sine function start repeating? 360°. ▫The sine and cosine functions are periodic, with a period of 360°. Synonyms: cyclic and cycle (or cycle length)

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Classwork / HW From Section 2-3: ▫On pp. 71-72, do Q1-Q10 and 1-23 odd. ▫Due Monday. Read Section 2-4 (pp. 73-77). Play an online game for 5-10 minutes. ▫Yes, a specific one: http://www.gimme5games.com/play-game/gimme-friction-baby http://www.gimme5games.com/play-game/gimme-friction-baby ▫Think about how it might be related to what we’re doing here. Write a brief paragraph in your journal explaining what you think.

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Clean-up / Reminders Pick up all trash / items. Push in chairs (at front and side tables). See you tomorrow!

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October 24, 2012 Unit Circle Warm-up! Your right arm is the initial side and your left arm is the terminal side. Ready? Let’s go! 360° - 3 6 0 ° -2π 2.

October 24, 2012 Unit Circle Warm-up! Your right arm is the initial side and your left arm is the terminal side. Ready? Let’s go! 360° - 3 6 0 ° -2π 2.

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