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4.3 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. Similarly, the domain of cosine is (−∞, ∞) and the range is [− 1 , 1].
4.3.1 Symmetries of sine and cosine
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 while a negative value of θ means we move clockwise. 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. 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. Reversing the direction of rotation reverses the sign of the y-value and so
sin(−θ) = − sin(θ).
Therefore the sine function is odd. Here is the graph of the sine function.
Figure 13. The graph of f (θ) = sin(θ).
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 .) The reasons for this are obvious: since 2π radians makes a complete revolution of the circle then
sin(θ + 2π) = sin(θ).
Some worked problems
- Describe the set of all the angles θ that satisfy the trig equation sin θ = −
√ 3
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, we see that θ =
4 π 3
and θ =
5 π 3
are angles whose sine is −
√ 3
But then since the sine function is periodic with period 2π we know that
θ =
4 π 3
5 π 3
where k is an integer, will also be solutions.
- Solve each of the following equations.
(a) cos x = 1 (b) 2 cos x = 1 (c) (2 cos x − 1)(cos x − 1) = 0.
Solutions.
(a) 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 { 2 πk : k ∈ Z}.
(b) Since cos π 3 = 12 then x = π 3 is a solution to 2 cos x = 1. So is x = − π 3. (Remember, f (x) = cos x is an even function!) Since the period of cosine is 2π then our set of all solutions is
{ π 3 + 2πk : k ∈ Z} ∪ {− π 3 + 2πk : k ∈ Z}.
(c) Any solution to (2 cos x − 1)(cos x − 1) = 0 is either a solution to cos x − 1 = 0 or a solution to 2 cos x − 1 = 0. We have already solved these equations in parts (a) and (b). All of the solutions to parts (a) and (b) are solutions to part (c). So our answer is
{ 2 πk : k ∈ Z} ∪ { π 3 + 2πk : k ∈ Z} ∪ {− π 3 + 2πk : k ∈ Z}.
4.3.2 Amplitude, period and phase shift
In practical applications many periodic functions are tranformations of the sine function. A transfor- mation of the sine function is often called a sine wave or a sinusoid. In general, sine waves will have form f (θ) = a sin(b(θ − c)) + d. (12)
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!):
- Shift right by c,
- Shrink horizontally by a factor of b,
- Expand vertically by a factor of a,
- Shift up by d.
Figure 14. The graph of f (x) = cos(θ).
Just as we did with sine waves, we may consider graphs of
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; here the function g(θ) has period p =
2 π |b|
, frequency f =
|b| 2 π
, amplitude
|a| 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 (or an examination of the definitions of cosines and sines on the unit circle) demonstrate that
the graph of cos(θ) is the graph of sin(θ) shifted to the left by
π 2
. Therefore
cos(θ) = sin(θ +
π 2
We could think of the cosine function as a sine wave with phase shift −
π 2
4.3.3 The symmetries of the six trig functions
Since the sine function is odd and the cosine function is even then
tan(−θ) =
sin(−θ) cos(−θ)
− sin(θ) cos(θ)
= − tan(θ)
and so the tangent function is odd. Here is a graph of the tangent function:
Figure 15. The graph of f (x) = tan(θ).
If the central angle θ gives the point P (x, y) on the unit circle then the tangent of θ is
y x
. The tangent
of θ + π will then be −y −x
and since the minus signs will cancel we see that
tan(θ + π) =
y x
= tan(θ).
So the tangent function has period p = π, not 2π! The reciprocals of cosine, sine and tangent with have the same “parity” (even/odd property) 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.
Worked problems with sine waves
- Describe the transformations necessary to change the graph of y = sin x into the graph of y = −5 sin(2(x − π 4 )) + 1 Solution. (These must be done in exactly this order. Any other order is incorrect.)
(a) Shift right by π 4. (b) Shrink horizontally by a factor of 2. (c) Expand vertically by a factor of 5 and reflect across the x-axis. (d) Shift up 1.
- Give the amplitude, period and phase shift for the sine wave y = −5 sin(2(x − π 4 )) + 1. Solution.
(a) In the previous problem we began by shifting right by π 4. This is the phase shift. (b) Then we shrunk the graph horizontally by a factor of 2 so the period is 22 π = π. (c) 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.
(a) What is the period of this function? (b) What is its amplitude? (c) What does the phase shift m = 4.5 say about the month of April? (d) Use this model to estimate the average high in January, April, July and October. How do those numbers compare with the data from the Weather Channel?
Solution. To transform the sine wave f (θ) = sin(θ) into the graph of H(m) = 15 sin( π 6 (m − 4 .5)) + 78 we must first shrink the graph horizontally by a factor of π/6 so that the wave has period 2 π π/ 6
= 12. 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.)
Notice that our model equation for Houston average high monthly temperatures, in the last problem, does not quite fit the Weather Channel data, but is certainly close. This is typical in scientific modeling
- we want a good first approximation and then we explore reasons that the model does not quite fit real life.
4.3.4 Other resources on sinusoidal functions
Many sources unfortunately define the trig functions on a triangle, as if all angles must lie between 0 and 180 degrees. In this course, trig functions are defined on the unit circle (where they exists “naturally”) and the triangle viewpoint is developed as a secondary concept, in a later lecture. Here we provide resources that describe trigonometry in terms of the unit circle.
In the free textbook, Precalculus, by Stitz and Zeager (version 3, July 2011, available at stitz-zeager.com) this material is covered in section 11.1.
In the free textbook, Precalculus, An Investigation of Functions, by Lippman and Rassmussen (Edition 1.3, available at www.opentextbookstore.com) this material is covered in section 6.1.
In the textbook by Ratti & McWaters, Precalculus, A Unit Circle Approach, 2nd ed., c. 2014 this material appears in section 4.4. In the textbook by Stewart, Precalculus, Mathematics for Calculus, 6th ed., c. 2012 (here at Amazon.com) this material appears in chapter 5.
There are some good online resources on the sine wave. Here are some I recommend:
- From Khan Academy are videos on the graph of the sine function.
- There is a short tutorial at mathopenref.com.
- Of course there is always a good article at Wikipedia.
- One should check out the cosine and sine waves of this Java applet at the German physics website walter-fendt.de. For other interesting physics applets, start at this webpage of Walter Fendt.
Homework. As class homework, please complete Worksheet 4.3, Sine Waves available through the class web- page.
