pencil, highlighter, calculator, notebook U8P2D1 pencil, highlighter, calculator, notebook Have out: Bellwork 1. Solve for all values of θ between 0 ≤ θ < 2π. Draw sketches. (Hint: rewrite the equation in terms of the trig ratio, then find the values for θ.) Sine is negative in QIII and QIV. 2 2 +1 solve for sin θ total:

Sine is negative in QIII and QIV. 30° 60° 1 reference angle: α = 60° +1 y θ +1 graphed angle x +1 α 1 +2 +1 triangle total:

Sine is negative in QIII and QIV. 30° 60° 1 reference angle: α = 60° y +1 graphed angle θ x +1 α 1 +2 +1 triangle total:

Intro to Graphing Trigonometric Functions Recall that any point on the unit circle has the coordinates cosθ sinθ y x (_____, _____) (0, 1) Record the coordinates for every multiple of 45°. (1, 0) 45° (–1, 0) (0, –1) Animation

Fill in the table with the decimal equivalent for sinθ. The Sine Function: Fill in the table with the decimal equivalent for sinθ.  0° 45° 90° 135° 180° 225° 270° 315° 360° sinθ 0.71 1 0.71 –0.71 –1 –0.71 Plot the points (θ, sinθ) on the graph below. 1 –1 sinθ θ 0° 45° 90° 135° 180° 225° 270° 315° 360° Animation

sinθ 1 θ 0° 45° 90° 135° 180° 225° 270° 315° 360° –1 QI QII QIII QIV height sinθ represents the ______ of the point (x, y) as the angle θ increases. positive In QI, sinθ is ________. sinθ reaches a maximum at θ = 90°. positive 180 In QII, sinθ is ________. sinθ = 0 at θ = ____°. negative In QIII, sinθ is ________. sinθ reaches a minimum at θ = ____°. 270 negative 360 In QIV, sinθ is ________. sinθ = 0 at θ = ____°. Animation

sinθ 1 θ 0° 45° 90° 135° 180° 225° 270° 315° 360° –1 QI QII QIII QIV From 0° ≤ θ ≤ 360°, θ completes one _________ of the unit circle. revolution From 0° ≤ θ ≤ 720°, θ completes ____ revolutions of the unit circle. two Animation

From 0° ≤ θ ≤ 360°, is one cycle, or _______. Practice # 1: Plot (θ, sin θ) for 0° ≤ θ ≤ 720°. Label all intercepts, maximums, and minimums. sinθ (90°, 1) (450°, 1) 1 (720°, 0) (0°, 0) (360°, 0) (540°, 0) (180°, 0) θ 0° 90° 180° 270° 360° 450° 540° 630° 720° –1 (270°, –1) (630°, –1) . period From 0° ≤ θ ≤ 360°, is one cycle, or _______. periods From 0° ≤ θ ≤ 720° is two _______ (cycles) of f(θ) = sin θ. We usually write the sine function as y = sin x.

Practice # 2: Sketch 3 periods of y = sin x. (360°)(3) = 1080° y 1 x 0° 180° 360° 540° 720° 900° 1080° –1 Scale the axes first. Be consistent. Plot the minimum and maximum points as well as any intercepts. Graph the curve.

Recall how y = x2 differs from y = –x2. How do you think y = –sin x will look? The graph will be reflected across the x–axis.

Practice # 3: Sketch 1 period of y = –sin x. Be sure to label the y–axis. 1 x 90° 180° 270° 360° –1

Recall how y = x2 differs from y = 2x2. How do you think y = 2sin x will look? The graph will be “stretched” vertically.

Practice # 4: Sketch 1 period of y = 2sin x. 90° 180° 270° 360° –1 –2

Given y = a sin x, |a| is the ________ (height) of the sine wave. amplitude If a < 0, then y = a sin x looks like: y a x 90° 180° 270° 360° –a When a < 0, the amplitude is never _______. The wave is just ___________. negative upside down Recall from 1st semester: this is true of the “stretch” factor for our graphs. The negative is not height. It indicates orientation.

Complete Practice # 5

Practice # 5: Sketch 1 period of the following functions Practice # 5: Sketch 1 period of the following functions. Be sure to scale and label the axes, and label all intercepts, maximums, and minimums. Use radian measure. a) y = 3 sin x y (90°, 3) 3 (180°, 0) (360°, 0) (0°, 0) x 90° 180° 270° 360° –3 (270°, –3)

b) y = –5 sin x y 5 x 90° 180° 270° 360° –5 (270°, 5) (360°, 0) (180°, 0) (0°, 0) x 90° 180° 270° 360° –5 (90°, –5)

c) y = sin x y (360°, 0) (180°, 0) (0°, 0) x 90° 180° 270° 360°

d) y = sin x y (180°, 0) (360°, 0) (0°, 0) x 90° 180° 270° 360°

Finish the assignment

old bellwork

pencil, highlighter, calculator 5/19/09 pencil, highlighter, calculator Have out: Bellwork a) If the given point is on the terminal side of θ and 0 ≤ θ < 2π, b) If θ = , find sinθ, cosθ, and tanθ. plot the point show θ find the radian measure of θ. y y x x

a) If the given point is on the terminal side of θ and 0 ≤ θ < 2π, plot the point show θ find the radian measure of θ. This is a 30°–60°–90°Δ. α = 30° since it is across from the shortest leg. y θ x α 1

find sinθ, cosθ, and tanθ. b) If θ = , y θ 1 x α = 60° 2 S Y R C X R T Y X sinθ = cosθ = tanθ =

Sine is negative in QIII and QIV. 30° 60° 1 reference angle: α = 60° +1 60˚ 60˚ +2 +2 total: +1 sketch +1 sketch