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