1. Graphing Sine and Cosine
Pre Calculus
Graphing Sine and Cosine

2. What you will learn How to graph sine and cosine functions.
How to translate sine and cosine functions (shift, left, right, vertical stretch, horizontal stretch)
How to use key points to “sketch” a graph.

3. Plan Discuss how to use the Unit Circle to help with graphing
Graphing Sine and Cosine and their translations

4. Fundamental Trigonometric Identities for
Cofunction Identities
sin  = cos(90  ) cos  = sin(90  )
sin  = cos (π/2  ) cos  = sin (π/2  )
tan  = cot(90  ) cot  = tan(90  )
tan  = cot (π/2  ) cot  = tan (π/2  )
sec  = csc(90  ) csc  = sec(90  )
sec  = csc (π/2  ) csc  = sec (π/2  )
Reciprocal Identities
sin  = 1/csc  cos  = 1/sec  tan  = 1/cot  cot  = 1/tan  sec  = 1/cos  csc  = 1/sin 
Quotient Identities
tan  = sin  /cos  cot  = cos  /sin 
Pythagorean Identities
sin2  + cos2  = 1 tan2  + 1 = sec2  cot2  + 1 = csc2 
Fundamental Trigonometric Identities for

5. Review of Even and Odd Functions
Cosine and secant functions are even
cos (-t) = cos t sec (-t) = sec t
Sine, cosecant, tangent and cotangent are odd
sin (-t) = -sin t csc (-t) = -csc t
tan (-t) = -tan t cot (-t) = -cot t

6. Graphing – Sine and Cosine

7. Key Things to Discuss Shape of the functions
Using the Unit Circle to help identify key points
Periodic Nature
Translations that are the same as other functions we have studied
Translations that are different than others we have studied
Using the calculator and correct interpretation of the calculator

8. Shape of Sine and Cosine
The unit circle: we imagined the real number line wrapped around the circle.
Each real number corresponded to a point (x, y) which we found to be the (cosine, sine) of the angle represented by the real number.
To graph the sine and cosine we can go back to the unit circle to find the ordered pairs for our graph.

9. Let’s convert some of these numbers to decimal form – start with cosine

10. Cosine Key Features to define the shape
Input: x
Angle
Output:
Cos x 1
max
π/6
.87
π/4
.71
π/3 .5
π/2
int
2π/3
-.5
3π/4
-.71
5π/6
-.87 π -1
min
7π/6
5π/4
4π/3
3π/2
5π/3
7π/4
11π/6 2π

11. Shape Calculator Mode: Radians
Window: Xmin 0 Xmax 2 Xscl /2 Ymin -2 Ymax Yscl .5
Y=cos x

12. Cosine For the function:
The angle is the input or independent variable and the cosine ratio is the output or dependent variable.
From a unit circle perspective, the input is the angle and the output is the “x” coordinate of the ordered pair.
Remember “coterminal angles” every 2π the values will repeat – this is called a periodic function
We will use the interval [0, 2π] as the reference period.

13. Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 -1
cos x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x
y = cos x
Cosine Function

14. Worksheet
Let’s graph it!

15. Now let’s look at Sine

16. Let’s convert some of these numbers to decimal form – start with sine

17. Sine Key Features to define the shape
Input: x
Angle
Output
sin x
int
π/6 .5
π/4
.71
π/3
.87
π/2 1
max
2π/3
3π/4
5π/6
Output: π
7π/6
-.5
5π/4
-.71
4π/3
-.87
3π/2 -1
min
5π/3
7π/4
11π/6 2π

18. Shape Calculator Mode: Radians
Window: Xmin 0 Xmax 2 Xscl /2 Ymin -2 Ymax Yscl .5
Y=sin x

19. Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. -1 1
sin x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x
y = sin x
Sine Function

20. Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range over an x-interval of
6. The cycle repeats itself indefinitely in both directions of the x-axis.
Properties of Sine and Cosine Functions

21. Transformations – A look back
Let’s go back to the quadratic equation in graphing form: y = a(x – h)2 + k
If a < 0: reflection across the x axis
|a| > 1: stretch; and |a| < 1: shrink
(h, k) was the vertex (locator point)
h gave us the horizontal shift
k gave us the vertical shift

22. Transforming the Cosine (or Sine)
y = a (cos (bx – c)) + d Let’s look at a |a| is called the amplitude, like our other functions it is like a stretch If a < 0 it also causes a reflection across the x-axis Graph y = cos x and y = 3 cos x y = cos x and y = -3 cos x

23. Key Points
Amplitude – increase the output by a factor of the amplitude
Remember the amplitude is always positive so you have to apply any reflections
y = 3 cos x 1 -1
cos x x 3 -3
cos x x

24. Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the reference interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
max
x-int
min 3 -3
y = 3 cos x 2  x y x
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)
Example: y = 3 cos x

25. If |a| > 1, the amplitude stretches the graph vertically.
The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| < 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis. y x
y = 2 sin x
y = sin x
y = sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x
Amplitude

26. Transforming the Cosine (or Sine)
y = a (cos (bx – c)) + d Let’s look at d Just like in our other functions, d is the vertical shift, if d is positive, it goes up, if it is negative, it goes down. Graph y = cos x and y = cos x + 1 y = cos x and y = cos x – 2

27. Vertical Shift 1 -1
cos x x
y = cos x + 1 Begin with y = cos x Add d to the output to adjust the graph Then shift up one unit y x 2 1
cos x x

28. Transforming the Cosine (or Sine)
y = a (cos (bx – c)) + d Let’s look at c Just like in our other functions, c is the horizontal shift, if c is positive, it goes right, if it is negative, it goes left. If there is no b present, it is the same as other functions Graph: y = cos x and y = cos (x + π/2) y = cos x and y = cos (x - π/2)

29. Horizontal shift y = cos (x + π/2) Begin with y = cos x1 -1
cos x x
y = cos (x + π/2)
Begin with y = cos x
Now you must translate the input… the angle
Then shift left π/2 units y x 1 -1
cos x x

30. Transforming Cosine (or Sine)
y = a (cos (bx – c)) + d
Let’s look at b
Graph:
y =cos x and y =cos 2x (b = 2)
y =cos x and y =cos ½ x (b = ½ )
What happened?

31. Transforming the Period
b has an effect on the period (normal is 2π) If b > 1, the period is shorter, in other words, a complete cycle occurs in a shorter interval If b < 1, the period is longer or a cycle completes over an interval greater than 2π
To determine the new period = 2π/b

32. If b > 1, the graph of the function is shrunk horizontally.
The period of a function is the x interval needed for the function to complete one cycle.
For b  0, the period of y = a sin bx is
For b  0, the period of y = a cos bx is also
If b > 1, the graph of the function is shrunk horizontally. y x
period:
period: 2
If 0 < b < 1, the graph of the function is stretched horizontally. y x
period: 4
period: 2
Period of a Function

33. Summarizing …
Standard form of the equations: y = a (cos (bx – c)) + d y = a (sin (bx – c)) + d
“a” - |a| is called the amplitude, like our other functions it is like a stretch it affects “y” or the output
If a < 0 it also causes a reflection across the x- axis
“d” – vertical shift, it affects “y” or the output
“c” – horizontal shift, it affects “x” or “θ” or the input
“b” – period change (“squishes” or “stretches out” the graph
The combination of “b” and “c” has another effect that we will discuss next time.

34. Next class
We will get into the detail of transforming the period when both b and c are present
We will graph using a “key point” method
We will talk about how the calculator can help and how you need to be careful with setting windows.

35. Calculator Issues Window settings
Using your reference period to set your window
Setting your scale

36. Homework 24
Section 4.5, p. 307
3-21 odd, all