Chapter 6: 6.3 - Investigating the Sine Function
Date:____________________
Investigating the Sine Function
Introduction to the Unit Circle
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
θ
opp y
= =y
hyp 1
∴ sin θ = y
y
sin θ =
Here,
x
Let’s place this triangle on a coordinate grid….
1
θ
y
x
Assuming that the hypotenuse remains constant (length of 1), the sine function will be equivalent to the
y-value. As the angle increases (counter-clockwise) the circle traces out a circle. (see GSP demo)
The angle θ is said to be in standard position in the xy plane since the
angle vertex lies on the origin (0, 0).
The point P(x, y) is any point on the terminal arm of the unit circle as θ
goes from 0Ε to 360Ε. Here the radius (r) equals one.
We then define the 3 primary trig ratios as follows…
y
x
y
sin θ =
cosθ =
tan θ =
r
r
x
Note: Since r, x and y form a right triangle, r 2 = x 2 + y 2
** In this unit, we will focus on the sine ratio only. **
MCF 3M – Grade 11 U/C Math
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Chapter 6: 6.3 - Investigating the Sine Function
Remember, given sin θ =
Date:____________________
y
, if r = 1 then we are dealing with the UNIT CIRCLE and sin θ = y
r
Let’s investigate the signs (+ve/-ve) of the sine ratio in quadrants 1 to 4.
θ
0o
30o
60o
90o
120o
150o
180o
210o
240o
270o
300o
330o
360o
sin θ
Summary,
1st Quadrant
sin θ (+ve/-ve)
2nd Quadrant
3rd Quadrant
4th Quadrant
We’ll now graph the above table of values for the sine ratio.
1
0.8
0.6
0.4
0.2
50
100
150
200
250
300
350
-0.2
-0.4
-0.6
-0.8
-1
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… R = {
}
MCF 3M – Grade 11 U/C Math
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Chapter 6: 6.3 - Investigating the Sine Function
Date:____________________
Terminology:
9 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
9 Sinusoidal function: a type of periodic function created by transformations of f(x) = sin x
9 Equation of the axis: the equation of the horizontal line halfway between the maximum and the
minimum is determined by…
9 Amplitude: the distance from the function’s equation of the axis to either the maximum or the
minimum value
For the Sine function y = sin x …
¾ The period: ___________
1
0.8
f(x ) = sin(x )
¾ The maximum value: ___________
0.6
0.4
¾ The minimum value: ___________
0.2
50
¾ The Equation of the axis is:
___________
100
150
200
250
300
350
-0.2
-0.4
-0.6
¾ The amplitude is: ___________
MCF 3M – Grade 11 U/C Math
-0.8
-1
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Chapter 6: 6.3 - Investigating the Sine Function
Date:____________________
Let’s Examine Example 1 on page 337
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:
a) The period: ___________
b) The maximum height: ___________
c) The minimum height: ___________
d) The Equation of the axis is: ___________
e) The amplitude is: ___________
Class work / Homework:
pg 339 – 343 #1, 2, 3abc, 4, 6a, 8, 9
MCF 3M – Grade 11 U/C Math
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