Download Chapter 11 Trigonometric Graphs and Identities

Chapter 11 Trigonometric Graphs and Identities
11.1A Graphs of Sine and Cosine
Objectives:
F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
F.TF.5: Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.
F.BF.3: Identify the effect on the graph of repleacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative); find the value of k given the graphs…
For the Board: You will be able to graph trigonometric functions.
Bell Work 11.1:
Evaluate:
π
π
π
1. sin
2. cos
3. cos
6
2
3
Find the measure of the reference angle for each given angle.
5. 145°
6. 317°
4. sin
π
4
Anticipatory Set:
Periodic functions are functions that repeat exactly in regular intervals called cycles.
The length of the cycle is called its period.
Instruction:
The sine and cosine functions are examples of periodic functions.
The circumference of the unit circle is 2so the period of the sine and cosine function is 2.
Graphing Activity:
Example: Graph y = sin x using a table established from the unit circle.
Example: Graph y = cos x using a table established from the unit circle.
y = sin x
x
0
π/6
π/2
5π/6
π
7π/6
3π/2
11π/6
2π
y = cos x
y
0
½
1
½
0
-½
-1
-½
0
2
1
0
π/2
-1
-1
-2
π
3π/2
2π
x
0
π/3
π/2
2π/3
π
4π/3
3π/2
5π/3
2π
y
1
½
0
-½
-1
-½
0
½
1
2
1
0
-1
-2
π/2
π
3π/2
2π
The amplitude (always positive) of the sine and cosine functions is half of the difference between the
maximum and minimum values of the function.
1 – (-1) = 2/2 = 1
Characteristics of the Sine and Cosine Function
1. Basic ordered pairs:
Sine: (0, 0), (π/2, 1), (π, 0), (3π/2, -1), (2π, 0)
Cosine: (0, 1), (π/2, 0), (π, -1), (3π/2, 0), (2π, 1)
2. domain: {x|x is all real numbers}
3. range: {y|-1 ≤ y ≤ 1}
4. period 2
5. amplitude 1
The sine and cosine functions can be “stretched” or “compressed” vertically and horizontally.
y = a sin bx
y = a cos bx
Recall: a indicates a vertical stretch or compression, which changes the amplitude
b indicates a horizontal stretch or compression, which changes the period.
The amplitude is |a|.
π
The period is .
b
Guidelines for Sketching the Graphs of Sine and Cosine Using the 5 Point Method
2π
1. Determine the period of the function. period =
b
2. Set up a table of 5 ordered pairs
0
x
¼ period
½ period
¾ period
1 period
y
3.
4.
5.
6.
Find the y’s by using the unit circle.
Label the x and y axes with evenly distributed scales.
Plot the points and draw the curve.
Draw additional cycles as needed.
Examples:
a. Graph y = -2 sin 4x
period = 2π/4 = π/2
x
0
π/8
π/4
3π/8
π/2
y
0
-2
0
2
0
b. Graph y = ¾ cos ½ x
period = 2π/ ½ = 4π
x
2
/
-1
3π/2
-2
0
0
π
2π - ¾
3π 0
4π ¾
2
¾
1
0
y
π/8
2π
π/4
3π/8
π/2
1
0
3
-1
/
-23π/2
π
2π
3π
4π
π/2
2π
Graphing Activity:
Practice:
a. Graph y = 3 sin 2x
Period = 2π/2 = π
x
0
π/4
π/2
3π/4
π
y
0
3
0
-3
0
b. Graph y = ½ cos ¾ x
Period = 2π/ ¾ = 8π/3
3
0
/
3π/2
-3
π/4
π/2
3π/4
π
2π
x
0
2π/3
4π/3
2π
8π/3
y
½
0
-½
0
½
1
½
0
/
-½
3π/2
-1
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 759 prob. 4 – 6, 14 – 17.
2π/3 4π/3
2π
2π
8π/3