1. 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:

2. 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:

3. 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:

4. 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

5. 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

6. 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

7. 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

8. 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.

9. 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.

10. 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.

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

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

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

14. 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.

15. Complete Practice # 5

16. 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)

17. 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)

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

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

20. Finish the assignment

21. old bellwork

22. 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

23. 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

24. 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 =

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