The Sine Wave
The sine function has domain the set of all real numbers: (−∞, ∞) but
the range is just [−1, 1] since all y-coordinates on the unit circle must be
between −1 and 1.
Elementary Functions
Part 4, Trigonometry
Lecture 4.3a, The Sine Wave
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU)
Elementary Functions
2013
1 / 29
Cosine
Smith (SHSU)
Elementary Functions
2013
2 / 29
Symmetries of sine and cosine
Similarly, the domain of cosine is (−∞, ∞) and the range is [−1, 1].
Let’s consider the definition of sine and cosine on the unit circle and ask
about symmetries. Are either of these functions even? odd?
We assume that a positive angle θ involves a counterclockwise rotation
starting at the point P (1, 0) and moving above the x-axis, while a
negative value of θ means we move clockwise, starting at the point P (1, 0)
and moving below the x-axis,
Is it clear that moving clockwise instead of counterclockwise does not
change the sign of the x-value of the point P (x, y)?
That is, for any angle θ,
cos(−θ) = cos(θ)
and so cosine is an even function.
Smith (SHSU)
Elementary Functions
2013
3 / 29
Smith (SHSU)
Elementary Functions
2013
4 / 29
Symmetries of sine and cosine
Central Angles and Arcs
Here is the graph of the sine function. Notice the rotational symmetry
about the origin.
But the sine function has much more symmetry than just rotational
symmetry about the origin. It is in fact periodic with period 2π (≈ 6.28.)
Since 2π radians makes a complete revolution of the circle then
However, if we begin to move clockwise around the origin, beginning on
the x-axis at (1, 0) then the y-value of the point P (x, y) immediately
becomes negative instead of positive.
sin(θ + 2π) = sin(θ).
Reversing the direction of rotation reverses the sign of the y-value and so
sin(−θ) = − sin(θ).
Therefore the sine function is odd.
Smith (SHSU)
Elementary Functions
2013
Smith (SHSU)
5 / 29
Some worked problems
Elementary Functions
2013
6 / 29
The Sine Wave
Solve each of the following equations.
1
Describe the
√ set of all the angles θ that satisfy the trig equation
sin θ = − 23 .
Solution. If the sine of an angle is negative then it must be in the
third of fourth quadrants. From our knowledge of 30-60-90 triangles,
5π
4π
(240◦ ) and θ =
(300◦ ) are angles whose sine
we see that θ =
3
3
√
is − 23 .
Since the sine function is periodic these are not the only solutions to
this equation! Since the sine function is periodic with period 2π we
know that
4π
5π
θ=
+ 2πk and θ =
+ 2πk,
3
3
(where k is an integer) will also be solutions.
Smith (SHSU)
Elementary Functions
2013
7 / 29
1
cos x = 1
2
2 cos x = 1
3
(2 cos x − 1)(cos x − 1) = 0.
Solutions.
1
2
Since cos(0) = 1 then cos x = 1 means that x is either 0 or 0 plus
some multiple of 2π. We can write this all in the form 0 + 2πk (where
k is an integer) or
{2πk : k ∈ Z}.
Since cos π3 = 12 then x = π3 is a solution to 2 cos x = 1. So also is
x = − π3 . (Remember, f (x) = cos x is an even function!) Since the
period of cosine is 2π then our set of all solutions is
{ π3 + 2πk : k ∈ Z} ∪ {− π3 + 2πk : k ∈ Z}.
Smith (SHSU)
Elementary Functions
2013
8 / 29
The Sine Wave
Amplitude, period and phase shift
In practical applications many periodic functions are transformations of the
sine function. A transformation of the sine function is often called a sine
wave or a sinusoid. In general, sine waves will have form
3
Solve the equation (2 cos x − 1)(cos x − 1) = 0.
f (θ) = a sin(b(θ − c)) + d.
Solution. Any solution to (2 cos x − 1)(cos x − 1) = 0 is either a
solution to 2 cos x − 1 = 0 or a solution to cos x − 1 = 0.
We have already solved these equations in the previous two problems.
All of the solutions the previous two problems are solutions to this
problem. So our answer is
{2πk : k ∈ Z} ∪ { π3 + 2πk : k ∈ Z} ∪ {− π3 + 2πk : k ∈ Z}.
Smith (SHSU)
Elementary Functions
2013
9 / 29
Period of a sine wave
(1)
From our earlier discussion of transformations, we see that one can
transform the graph of sin(θ) into the graph of f (θ) = a sin(b(θ + c)) + d
by the following steps (in this order!):
1 Shift right by c,
2 Shrink horizontally by a factor of b (about the line x = c),
3 Expand vertically by a factor of a
4 Shift up by d.
Some of these translations are associated with particular terms. We revisit
the concept of period and introduce new terms frequency, amplitude and
phase shift.
Smith (SHSU)
Elementary Functions
2013
10 / 29
Frequency of a sine wave
The period of a sine wave tells us how many units of the input variable are
required before the function repeats.
Since the sine function has period 2π then the sine wave given by the
2π
function f (θ) = a sin(b(θ − c)) + d will have period
.
|b|
(We use an absolute value sign here since we want the period to be
positive and it is possible that b is negative.)
The frequency a sine wave is the number of times the wave repeats within
a single unit of the input variable θ; this is the reciprocal of the period.
Thus the frequency of the standard sine wave sin(x) is
frequency of f (θ) = a sin(b(θ − c)) + d is
Smith (SHSU)
Elementary Functions
2013
11 / 29
Smith (SHSU)
|b|
.
2π
Elementary Functions
1
and so the
2π
2013
12 / 29
Frequency
Amplitude of a sine wave
Electronic transmissions involve the sine wave. The frequency of the
transmission represents the number of copies of the sine wave which occur
within a single unit of time (often one second.)
The height of the standard sine wave oscillates between a maximum of 1
and minimum of −1.
For example, an electromagnetic wave with frequency 4.3 × 1014 oscillates
430, 000, 000, 000, 000 (430 million million) times in one second and is
perceived by our eyes as the color red.
Scientists often use the term “hertz” to represent “cycles per second” and
so we say that frequency of red light is 4.3 × 1014 hertz.
A light wave of lower frequency will not be visible to our eyes; waves of
higher frequency will show up as orange, yellow, green, and so on.
Smith (SHSU)
Elementary Functions
2013
13 / 29
Phase shift of a sine wave
If we consider the midpoint of this wave, then the wave rises 1 unit above
and then drops 1 unit below this midpoint.
This variation from the “average” height is the amplitude of the sine
wave. For the standard sine wave the amplitude is 1.
The amplitude of the sine wave f (θ) = a sin(b(θ − c)) + d is just |a|.
(Again, we use absolute value because we want the amplitude to be
positive.)
Smith (SHSU)
14 / 29
In the next presentation, we will continue to look further at sine waves
(sinusoids) including the cosine function.
A sine wave might be shifted to the right by an amount c; this is the
phase shift of the sine wave f (θ) = a sin(b(θ − c)) + d.
(End)
Note that the phase shift can be negative. A negative phase shift means
that the graph of sin θ is being shifted by a certain amount to the left.
Elementary Functions
2013
The sine wave
The graph of the standard sine wave sin(θ) passes through the origin
(0, 0).
Smith (SHSU)
Elementary Functions
2013
15 / 29
Smith (SHSU)
Elementary Functions
2013
16 / 29
The graph of cosine
.
The graph of cosine function has a very similar wave pattern to that of
sine.
Here is a graph of the cosine function.
Elementary Functions
Part 4, Trigonometry
Lecture 4.3b, The Cosine and other Sinusoids
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU)
Elementary Functions
2013
17 / 29
Smith (SHSU)
Elementary Functions
2013
Central Angles and Arcs
The symmetries of the six trig functions
Just as we did with sine waves, we may consider graphs of
Since the sine function is odd and the cosine function is even then
sin(−θ)
− sin(θ)
tan(−θ) =
=
= − tan(θ)
cos(−θ)
cos(θ)
and so the tangent function is odd. Here is a graph of the tangent
function:
g(θ) = a cos(b(θ − c)) + d.
There is no significant difference in meaning for the period, frequency,
amplitude or phase shift when discussing the cosine function.
The function g(θ) has period p =
18 / 29
2π
|b|
, frequency f =
, amplitude |a|
|b|
2π
and phase shift c.
We tend to concentrate on the sine wave and ignore the cosine function.
This is merely because the graph of cosine function is really a shift of the
graph of sine! A careful examination of the graphs of these functions
demonstrate that the graph of cos(θ) is the graph of sin(θ) shifted to the
π
left by .
2
π
cos(θ) = sin(θ + ).
2
π
We could
think of the cosine function
as a sine wave with phase shift
−19 /.29
Smith (SHSU)
Elementary Functions
2013
2
Smith (SHSU)
Elementary Functions
2013
20 / 29
The symmetries of the six trig functions
Symmetries of six trig functions
If the central angle θ gives the point P (x, y) on the unit circle then the
−y
y
and since the
tangent of θ is . The tangent of θ + π will then be
−x
x
minus signs will cancel
y
tan(θ + π) = = tan(θ).
x
So the tangent function has period p = π, not 2π!
The reciprocals of cosine, sine and tangent have the same “parity”
(even/odd-ness) as the original function.
So the secant function is even while cosecant and cotangent are both
odd.
Just like cosine and sine, the secant and cosecant functions have period
2π.
The cotangent function, like the tangent function, has period π.
Smith (SHSU)
Elementary Functions
2013
Smith (SHSU)
21 / 29
Worked problems with sine waves
2013
22 / 29
The Sine Wave
2
1
Elementary Functions
Describe the transformations necessary to change the graph of
y = sin x into the graph of y = −5 sin(2(x − π4 )) + 1
Give the amplitude, period and phase shift for the sine wave
y = −5 sin(2(x − π4 )) + 1.
Solution.
Solution. (These must be done in exactly this order. Any other order
is incorrect.)
Shift right by π4 .
Shrink horizontally by a factor of 2 (about the line x = π4 ).
3 Expand vertically by a factor of 5 and reflect across the x-axis.
4 Shift up 1.
1
2
Smith (SHSU)
Elementary Functions
2013
23 / 29
In the previous problem we began by shifting right by π4 . This is the
phase shift.
2 Then we shrunk the graph horizontally by a factor of 2 so the period is
2π
2 = π.
3 Then we stretched the graph vertically by a factor of 5 and turned it
over. The amplitude should always be positive (it represents a
deviation from the mean) and so the amplitude is 5.
1
Smith (SHSU)
Elementary Functions
2013
24 / 29
Examples of Sine Waves
3
The Sine Wave
A person’s blood pressure follows a sine wave corresponding to the
beats of the heart. A particular individual’s blood pressure at time t
(measured in minutes) is
p(t) = 20 sin(160πt) + 100.
What does this tell us about the person’s heart rate and blood
pressure?
Solution. To transform the sine wave f (θ) = sin(θ) into the graph of
p(t) = 20 sin(160πt) + 100 we first shrink the graph horizontally by a
2π
1
factor of 160π so that the wave has period 160π
= 80
.
Then stretch the graph vertically by a factor of 20 and then shift it up
100.
1
This means that a single heartbeat occurs in 80
th of a minute and
that the frequency is 80 beats per minute.
The amplitude of this sine wave is 20; the blood pressure varies from
a maximum of 100 + 20 = 120 to a minimum of 100 − 20 = 80. So
the person’s blood pressure is 120/80.
Smith (SHSU)
2013
25 / 29
The
phase shift is zero. Elementary Functions
The Sine Wave
4
Data from The Weather Channel summarizes Houston weather
averages.
One approximation to the average highs in Houston is the equation
H(m) = 15 sin( π6 (m − 4.5)) + 78
where m = 1 represents the month of January, etc. (See orange curve
below.)
Smith (SHSU)
Elementary Functions
2013
26 / 29
The Sine Wave
Given the equation,
H(m) =
15 sin( π6 (m
H(m) = 15 sin( π6 (m − 4.5)) + 78
− 4.5)) + 78
answer the following questions.
1 What is the period of this function?
2 What is its amplitude?
3 What does the phase shift m = 4.5 say about the month of April?
4 Use this model to estimate the average high in January, April, July
and October. How do those numbers compare with our chart?
Solution. To transform the sine wave f (θ) = sin(θ) into the graph of
H(m) = 15 sin( π6 (m − 4.5)) + 78 we shift the sine wave right by 4.5 and
then shrink the graph horizontally by a factor of π/6. Since we shrunk the
2π
= 12.
graph horizontally by a factor of π/6, the period is now
π/6
Not surprisingly, this tells us that the graph repeats every 12 months.
We then stretch the graph vertically by a factor of 15 (so that the
amplitude is 15) and then shift it up 78 so that if varies from a high of 93
to a low of 63.
The phase shift of m = 4.5 tells us that the month of April is close to the
annual average; it is 3 months after the lowest temperatures (in January)
and 3 months before the highest temperatures (in July/August.)
Smith (SHSU)
Elementary Functions
2013
27 / 29
Notice that our model equation for Houston average high monthly
temperatures,
does not quite fitElementary
the Weather
Smith (SHSU)
Functions Channel data, but is
2013
28 / 29
The Sine Wave
In the next presentation, we will look trigonometry on right triangles.
(End)
Smith (SHSU)
Elementary Functions
2013
29 / 29