MCF3M – Chapter 6: Sinusoidal Functions
Date:
6.3 Investigating the Sine Function
Goal: Identify properties of a specific type of periodic function called a sinusoidal function
So far, we’ve used the sine ratio with acute triangles. We’ll now examine the possibility of
determining the sine ratio of angles that are greater than 90 degrees.
Let’s consider a right angled triangle where the hypotenuse has a value of one (1).
1
opp y
=
hyp 1
∴ sin θ = y
sin θ =
y
For this triangle,
θ
x
Let’s place this triangle on a coordinate grid….
P(x, y)
1
θ
y
x
Assuming that the hypotenuse remains constant (length of 1), the sine value will be equivalent to the
y-value of the point P. As the angle increases (counter-clockwise) the line traces out a circle.
Let’s investigate the sign (positive or negative) of the sine ratio for angles in quadrants 1 to 4.
Complete the table using a calculator…
o
o
o
o
o
θ
0
30
60
90
120
150
180
210
240
270
300
330
360
sin θ
0.5
0
-0.5
-.866
-0.5
-.866
-0.5
0
0
0.5
0.866
1
o
0.866
o
o
o
o
o
o
Summary
sin θ (+ve/-ve)
1st Quadrant
(0o – 90o)
2nd Quadrant
(90o – 180o)
3rd Quadrant
(180o – 270o)
4th Quadrant
(270o – 360o)
Positive
Positive
Negative
Negative
We’ll now graph the above table of values for the sine ratio.
o
MCF3M – Chapter 6: Sinusoidal Functions
Date:
Graph of the Sine Function f(x) = sin θ
Yes
It passes the vertical line test
Is the above curve a function? __________
Why or why not? ______________________________
__________________________________________________________________________________
From the graph we can determine the domain and range of the sine function.
The Domain for the sine function is… D = {
The Range for the sine function is…
x∈R
}
R = { y ∈ R | -1 ≤ y ≤ 1 }
MCF3M – Chapter 6: Sinusoidal Functions
Date:
Terminology
¾ Sine Function: a sine function is the graph of f(x) = sin x, where x is an angle measured in
degrees; it is a periodic function
¾ Sinusoidal Function: a type of periodic function created by transformations of f(x) = sin x
¾ Equation of the Axis: the equation of the horizontal line
halfway between the maximum and the minimum. It is
determined by the formula…
y=
(maximum + minimum)
2
¾ Amplitude: the distance from the function’s equation of the
axis to either the maximum or the minimum value
|y| Æ always positive value
Characteristics of the Sine Function
y = sin x
360
¾ The period: ___________
o
1
¾ The maximum value: ___________
-1
¾ The minimum value: ___________
y=0
¾ The Equation of the Axis is: ___________
1
¾ The amplitude is: ___________
x∈R
¾ The domain is: ________________________
y ∈ R | -1 ≤ y ≤ 1
¾ The range is: ________________________
(0 , 0)
¾ 5 key points are: ___________
o
(90 , 1)
___________
o
(180 , 0)
___________
o
(270 , -1)
___________
Characteristics of Sinusoidal Functions
periodic
¾ Must be ___________________
shape
¾ Must have the same ____________________
as the sine function y = sin x
o
(360 , 0)
___________
o
MCF3M – Chapter 6: Sinusoidal Functions
Date:
Let’s Examine Example 1 on page 338
Example
Sarah is sitting in an inner tube in a wave pool. The depth of the water below her in terms of time
can be represented by the graph shown below.
Determine each of the following:
3 sec
a) The period: ___________
2.4
b) The maximum height: ___________
1.6
c) The minimum height: ___________
y = (2.4 + 1.6) ÷ 2 = 2
d) The Equation of the Axis is: ___________
– 2 = 0.4
__________
e) The amplitude is: _2.4
Sinusoidal
f) Is this a sine function or a sinusoidal function? ___________________
Explain.
It is sinusoidal since it has the same shape as the sine function BUT
_________________________________________________________________________
the amplitude is less than one and the equation of the axis ≠ 1.
_________________________________________________________________________
Classwork / Homework:
Pg 339 #1, 2, 3abc, 4, 6a, 8, 9