Derivative of the Sine Function By James Nickel, B.A., B.Th., B.Miss., M.A. .
The Sine Function - Katy Independent School...
Transcript of The Sine Function - Katy Independent School...
The sine function, y = sin θ, matches the measure
θ of an angle in standard position with the y-
coordinate of a point on the unit circle.
This point is where the terminal side of the angle
intersects the unit circle.
sin0π 0π
sin 12
sinπ 03π
sin 12
sin 2π 0
The graph on the last slide is called the parent
function of the sine function.
This means that:
1. it has not been stretched or compressed,
2. it has not been reflected over an axis,
3. it has not been moved up/down or right/left.
Today we will look at the first two
transformations.
The sine curve is a periodic function.
A periodic function repeats a pattern of y-values at
regular intervals.
One complete pattern is a cycle.
The period of a function is the horizontal length
of one cycle.
The period of the parent function sine curve is 2π.
Finding the Period of y = sin bθ
The period of a sine curve in the form y = sin bθ
is
b is the number of cycles that can be drawn from
0π and 2π.
2πp
b
Example
Find the period of the following sine curves and
tell how many cycles will be drawn from 0 to 2π:
a. y = sin 4θ
2π
4p
π
2p 4 cycles
Amplitude of a Sine Curve
The amplitude of a periodic function is half the
difference between the maximum and minimum
values of the function.
The amplitude of the parent function sine curve
is 1.
To find the equation for the sine curve, you must
find b. Remember b is the number of cycles from
0π to 2π.
The equation is y = a sin bθ.
3b
2sin3θy
Reflection Across the x-axis
If a in the equation y = a sin bθ is negative, then
the sine curve is reflected across the x-axis.
Example
Sketch one cycle of the graph of the sine curve.
a. y = 3 sin 2θ
1. The amplitude of the curve is 3. Mark this
on the y-axis.
2. The period is
2π
2p
πp
3. Divide the period into fourths to find the
hash marks for the graph.
4. The pattern for a sine curve is
zero-maximum-zero-minimum-zero
π
4
b.
1. What is the amplitude of the sine curve?
amplitude = 2
2. What is the period?
period = 6π
θ2sin
3y
3. What would you divide x-axis into?
4. What is the pattern for a sine curve?
zero-maximum-zero-minimum-zero
6π
4
3π
2
c. y = -4 sin 3θ
1. What is the amplitude of the sine curve?
amplitude = 4
2. What is the period?
2π
3p
3. What would you divide x-axis into?
4. What is the pattern for a sine curve?
Since there is a negative sign at the
beginning of the equation, we reflect the
graph over the x-axis.
zero-minimum-zero-maximum-zero
2π
3
4
π
6