1. Quiz 4-3 1.Find a positive and negative co-terminal angle with: co-terminal angle with: 2.Find a positive and negative co-terminal angle with: co-terminal angle with: 100º 3. A ray passes through the point (4,5). Find 4. Find tan 300º without a calculator.

2. Mini-TEST 1. On a unit circle, label all of the “nice angles” that come from the 45-45-90 and 30-60-90 right triangles in both degrees and radians. 2. Label the x and y coordinates for each of the points on the circle where the terminal side of the above angles intersect the circle. Without using a calculator find the following: 3. sec 45 4. Cos 225 5. Csc 3π/2 6. Tan π

3. 4.4 Graphs of Sine and Cosine: Sinusoids

4. What you’ll learn:  The Basic Waves Revisited  Sinusoids and Transformations  Modeling Periodic Behavior with Sinusoids … and why Sine and cosine gain added significance when used to model waves and periodic behavior.

5. Graph of the Sine Function f (x) = sin x radians Think of a dot traveling around the circle to the right. At the exact same time, a corresponding dot is traveling along the x-axis to the left.

6. Graph of the Sine Function f (x) = sin x radians Think of ‘x’ as either (1) the central angle or (2) the distance around the circle starting at the + x-axis.  if radius =1 then angle measure in radians = arc length.  if radius =1 then angle measure in radians = arc length.

7. Graph of the Sine Function: f (x) = sin x radians Think of sin x as the distance above (or below) the x-axis determined by ‘t’ (the distance around the circle). determined by ‘t’ (the distance around the circle).

8. Sinusoid Your turn: 1. Domain = ? 2. Range = ? 3. Continuous? 4. Symmetry? 5. bounded? 6. Vertical/horizontal asymptotes? 7. End behavior?

9. Sinusoid

10. Sinusoid Vertical stretch/shrink If a < 0: reflection across x-axis Horizontal stretch/shrink by factor of: If b < 0: reflection across y-axis Horizontal translation (phase shift) Vertical translation

11. Careful, Careful, Careful !! f(x – 1) is f(x) shifted to the Right by 1 Let’s say we start with: Then: f(x – 1) = sin(2(x -1)) (Replace ‘x’ with ‘x – 1’) Distributive property: f(x – 1) = sin(2x – 2) IMPORTANT: horizontal shifts are ‘c ÷ b’ !!!! Horizontal stretches affect both the variable along the x-axis AND the phase shift. along the x-axis AND the phase shift.

12. Vertical stretch/shrink: factor of 3 If a < 0: reflection across x-axis: none Horizontal stretch/shrink by factor of:

13. Vertical stretch/shrink: factor of 3 If a < 0: reflection across x-axis: none Horizontal stretch/shrink by factor of: If b < 0: reflection across y-axis: none Horizontal translation: left by Vertical translation: up by 5 units

14. Your turn: describe the transformations of f(x) 8. Vertical stretch/shrink 9. Horizontal stretch/shrink 10. Horizontal translation (phase shift) 11. Vertical translation

15. Sinusoid 1.Amplitude: ( ½ of peak to peak distance) = ( ½ of peak to peak distance) = 2. Period: length of horizontal axis encompassing one complete cycle = one complete cycle = 3. Frequency = Frequency = 1/period

16. 1.Amplitude: ( ½ of peak to peak distance) = ( ½ of peak to peak distance) = 2. Period: length of horizontal axis encompassing one complete cycle = one complete cycle = 3. Frequency =

17. Your turn: describe the transformations of f(x) 12. Amplitude = ? 13. Period = ? 14. Frequency = ?

18. Sine vs. Cosine Function Cos x = sin x shifted to the left by

19. Construct a Sinusoid Period = Amplitude = 6 Passes thru: (2, 0) Solution: 1. Find ‘b’ (coefficient of ‘x’) 2. Amplitude is easy: a = ± 6 Either works, use +10 Either works, use +6

20. Construct a Sinusoid Period = Amplitude = 6 Passes thru: (2, 0) 3. Find ‘c’: does everything but pass through (2, 0). pass through (2, 0). It does pass through (0, 0). We just need a phase shift of +2. After distributive property:

21. Construct a Sinusoid Period = Amplitude = 2 Passes thru: (0, 7) Solution: 1. Find ‘b’ (coefficient of ‘x’) 2. Amplitude is easy: a = ± 2 Either works, use +3 Either works, use +2

22. Construct a Sinusoid Period = Amplitude = 2 Passes thru: (0, 7) 3. Find ‘c’: does everything but pass through (0, 7). pass through (0, 7). It does pass through (0, 0). We just need to shift up by 7.

23. Construct a Sinusoid Period = Amplitude = 4 Passes thru: (2, 7) Solution: 1. Find ‘b’ (coefficient of ‘x’) 2. Amplitude is easy: a = ± 4 Either works, use Either works, use +4

24. Construct a Sinusoid Period = Amplitude = 4 Passes thru: (2, 7) 3. Find ‘c’: does everything but pass through (2, 7). pass through (2, 7). It does pass through (0, 0). We just need to shift up by 7 and have a phase shift of 2 to the right. shift of 2 to the right. After distributive property:

25. HOMEWORK Section 4-4 Section 4-4

26. Another example f(x) = ? When: minimum value: y = 5 at x = 0 and maximum value: y = 25 at x = 32 and maximum value: y = 25 at x = 32 1. Find amplitude: Remember: the amplitude is half the “peak to peak” distance the “peak to peak” distance 2. Find period: Period is the horizontal distance to from minimum to maximum and back to minimum. minimum to maximum and back to minimum. Period = 64

27. Another example f(x) = ? When: minimum value: y = 5 at x = 0 and maximum value: y = 25 at x = 32 and maximum value: y = 25 at x = 32 4. Use a horizontal translation of either the sine or cosine function to match up (0, 5) sine or cosine function to match up (0, 5) Cosine has a maximum value at x = 0  Reflect cosine across x-axis to get a minimum value at x = 0. 5. Vertical translation to make minimum value = 5 minimum value = 5 Move up 15 units.

28. Another example f(x) = ? When: minimum value: y = 5 at x = 0 and maximum value: y = 25 at x = 32 and maximum value: y = 25 at x = 32 1. Find amplitude: 2. Find period: Period = 64 3. Find the coefficient of ‘x’: 4. Use a horizontal translation of either the sine or cosine function to match up (0, 5) sine or cosine function to match up (0, 5) 5. Vertical translation to make minimum value = 5 minimum value = 5