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.
Plan Discuss how to use the Unit Circle to help with graphing Graphing Sine and Cosine and their translations
Fundamental Trigonometric Identities 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 sin 2 + cos 2 = 1 tan 2 + 1 = sec 2 cot 2 + 1 = csc 2 Fundamental Trigonometric Identities for
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
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
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.
Let’s convert some of these numbers to decimal form – start with cosine
Cosine Key Features to define the shape Input: x Angle Output: Cos x 0 1 max π/6.87 π/4.71 π/3.5 π/2 0 int 2π/3 -.5 3π/4 -.71 5π/6 -.87 Input: x Angle Output: Cos x π min 7π/6 -.87 5π/4 -.71 4π/3 -.5 3π/2 0 int 5π/3.5 7π/4.71 11π/6.87 2π 1 max
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.
Cosine Function 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. 1001cos x 0x 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
Let’s convert some of these numbers to decimal form – start with sine
Sine Key Features to define the shape Input: x Angle Output sin x 0 int π/6.5 π/4.71 π/3.87 π/2 1 max 2π/3.87 3π/4.71 5π/6.5 Input: x Angle Output: sin x π 0 int 7π/6 -.5 5π/4 -.71 4π/3 -.87 3π/2 min 5π/3 -.87 7π/4 -.71 11π/6 -.5 2π 0 int
Sine Function 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. 0010sin x 0x 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
Properties of Sine and Cosine Functions 6. The cycle repeats itself indefinitely in both directions of the x-axis. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 1. The domain is the set of real numbers. 5. Each function cycles through all the values of the range over an x-interval of. 2. The range is the set of y values such that.
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
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
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 30-303cos x 0x 1001cos x 0x
y x Example: y = 3 cos x 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. maxx-intminx-intmax 30-303 y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)
Amplitude 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 = – 4 sin x reflection of y = 4 sin x y = 4 sin x y = 2 sin x y = sin x
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
Vertical Shift 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 1001cos x 0x 21012 0x
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 )
Horizontal shift 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 1001cos x 0x 1001cos x 0 x
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?
Transforming the Period b has an effect on the period (normal is 2 π ) 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 2 π To determine the new period = 2π/b
y x Period of a Function period: 2 period: 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 0 < b < 1, the graph of the function is stretched horizontally. If b > 1, the graph of the function is shrunk horizontally. y x period: 2 period: 4
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.
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.
Calculator Issues Window settings Using your reference period to set your window Setting your scale
Homework 24 Section 4.5, p. 307 3-21 odd, 23-26 all