The document provides steps to sketch the path of a rock launched from a catapult using trigonometric functions. It determines that the path will be a negative cosine curve. It then finds the parameters for the cosine equation f(x) = -90cos(x/50) + 90 based on the given maximum height and range of the catapult. Finally, it derives the equivalent sine equation f(x) = -90sin(x+22.5)+90.

1. The Crusades “ Siege of Jerusalem”

2. You have been assigned as the head engineer for one of the Catapults that should siege a well fortified city. Your catapult is 6 meters long. The enemy guards are able to attack within 50 meters of their walls. The catapult that you have is able to shoot rocks has a maximum range of 100 meters and is able to catapult rocks from 180 meters high.

3. A. Sketch a graph that shows the maximum range of the catapult The first step to solving any problem would be to look for the given… Maximum range of Catapult = 100 Maximum Height of Boulder = 180

4. Step 1: Sine or Cosine? The first thing that we need to know is if the graph we are looking for is a sine or a cosine graph. We are looking for the graph of the catapult’s projectile. Using the method of elimination + common sense we can conclude…

5. Step 1: Cosine or Sine? f(x) = sinx f(x) = -cosx f(x) = -sinx f(x) = cosx

6. Step 1: Cosine or Sine? Using common sense we can try to find the kind of graph to use… It can’t be a sine graph because a projectile launched from a catapult cannot go up then down… Also it can’t be a + cosine graph because that would mean that projectile goes underground. This leaves the negative cosine graph for us to use…

7. Step 1: Cosine or Sine? f(x) = sinx f(x) = -cosx f(x) = -sinx f(x) = cosx

8. Step 2: Determine the Parameters Now that we know that the graph is a negative cosine graph… We should now use the formula: f(x) = A cos B ( x –C ) + D A is the Amplitude B is the determining factor for the period C is the horizontal shift or phase shift D is the vertical shift; sinusoidal axis ; average value of the function

9. Step 2: Determine the Parameters Looking back to the definitions of these terms… Amplitude is the distance of the maximum and minimum points from the sinusoidal axis Period is the length of each graph, cycle Horizontal Shift is the origin’s horizontal distance from point (0,0) Vertical Shift is the vertical distance of the graph’s origin from point (0,0)

10. Step 3: Sketch the Graph

11. Step 3: Sketch the Graph This image shows the catapult’s maximum height and maximum firing range

12. Step 3: Sketch the Graph The Catapult shoots its projectiles in a parabolic manner so the graph would look like this….

13. Step 3: Sketch the Graph The graph of the projectile’s path would be like this…

14. B. Show the equations (one for cosine and one for sine) for the rocks path. When doing equations for functions we can use DBAC to help us…. f(x) = A cos B ( x –C ) + D DBAC just means that we should follow the order of the letters… Find D (sinusoidal axis) first, then look for A(Amplitude), B (Determines the Period), then C(Horizontal Shift).

15. Step 1: Find Parameter D Parameter D, also known as the Vertical shift can also be defined as the sinusoidal axis or the average value of the function… This means that be getting the average value of the maximum and minimum of the graph we will have the average value or the sinusoidal axis…

16. Step 1: Find Parameter D Average Value of the Function = Max Value + Min Value 2 In this given situation or minimum is at 0 (Ground Level) and our max is at 180 meters…

17. Step 1: Find Parameter D Average Value of the Function = 0 + 180 2 Average Value of the Function = 90 Therefore Parameter D is equal to 90

18. B. Show the equations (one for cosine and one for sine) for the rocks path. f(x) = A cos B ( x –C ) + 90 DBAC just means that we should follow the order of the letters… Find D (sinusoidal axis) first, then look for A(Amplitude), B (Determines the Period), then C(Horizontal Shift).

19. Step 2: Find Parameter B Now it’s time to look for parameter B. Parameter B DETERMINES THE PERIOD …It is NOT the Period… How do you find the period? Period = 2∏ B Period 2∏ B =

20. Step 2: Find Parameter B Look back at the graph. How long is one period of our – cos x graph? This is one period… Period = 100

21. B . Show the equations (one for cosine and one for sine) for the rocks path. f(x) = A cos ( x –C ) + 90 DBAC just means that we should follow the order of the letters… Find D (sinusoidal axis) first,), B (Determines the Period), then look for A(Amplitude then C(Horizontal Shift). π 50

22. Step 3 : Find Parameter A Amplitude is the distance of the maximum and minimum points from the sinusoidal axis

23. Step 3 : Find Parameter A The amplitude can be figured out by subtracting the maximum or the minimum value to the average value of the graph… A = 180 - 90 A = 90 - 0 A = 90 A = 90 A= Max Value – Sinusoidal Axis A= Sinusoidal Axis – Min Value

24. B . Show the equations (one for cosine and one for sine) for the rocks path. f(x)= -90 cos (x –C) + 90 DBAC just means that we should follow the order of the letters… Find D (sinusoidal axis) first,), B (Determines the Period), then look for A(Amplitude then C(Horizontal Shift). π 50

25. Step 4: Find Parameter C Parameter C is the Horizontal Shift. Since Graph starts from Origin, therefore the value of Parameter is 0

26. B . Show the equations (one for cosine and one for sine) for the rocks path. f(x)= -90 cos ( x – 0 )+ 90 DBAC just means that we should follow the order of the letters… Find D (sinusoidal axis) first,), B (Determines the Period), then look for A(Amplitude then C(Horizontal Shift). π 50

27. Show the equations (one for cosine and one for sine) for the rocks path. f (x) = - 90 cos x+ 90 B . To get the equivalent sine function of that one above, we’d just have to shift the graph to where the function intersects the sinusoidal axis… Tip: When getting equivalent functions the only thing that changes would be Parameter C or the Phase Shift… π 50

28. Step 5: Find the Sine Equation The arrows show where the sine function can start…

29. Step 5: Find the Sine Equation The light blue boxes shows the negative and positive sine functions…

30. Step 5: Find the Sine Equation By looking for the x- coordinates when the value of y is equal to the sinusoidal axis, we can find where the graph starts, which is equal to the Phase Shift

31. Step 5: Find the Sine Equation f (x) = -90 sin( x + 22.5 )+90 f (x) = 90 cos ( x - 50)+ 90 π 50 π 50 f (x) = 90 sin( x – 22.5 )+90 π 50 f (x) = - 90 cos x+ 90 π 50