Show that the function is a sinusoid by rewriting it

In summary, the conversation discusses a student's attempt at solving a trigonometry problem involving sinusoid functions. The student presents their solution and asks for help in understanding why their answer differs from the answer given in the book. Through further discussion and clarification, it is discovered that the student made an error in their calculations and the book's answer is correct. The conversation also touches on the concept of unique solutions in trigonometry problems and the importance of paying attention to signs and quadrants when solving.
  • #1
Vital
108
4

Homework Statement


Hello!

I am doing exercises on sinusoid functions from the beginning of Trigonometry.
I hoped I understood the topic, but it seems not quite, because I don't get the results authors show as examples for one of possible answers, as there can be a few answers to the same exercise. I will be grateful for your help and explanation on what I am doing wrong.

Here is the task:
Show that the function is a sinusoid by rewriting it in the forms C(x) = A cos(ωx + φ) + B and S(x) = A sin(ωx + φ) + B for ω > 0.

Homework Equations


I will show one example, and, maybe, later add more, if it is necessary.
f(x) = 3√3 sin(3x) - 3cos(3x)

Here is the answer from the book, and below I post how I tried to solve the task:
answer from the book f(x) = 3√3 sin(3x) - 3cos(3x) = 6 sin( 3x + (11π/6) ) = 6cos( 3x + (4π/3) )

The Attempt at a Solution


My solution:
f(x) = 3√3 sin(3x) - 3cos(3x)
(1) rewrite the expression in the sinusoid form:
f(x) = A sin(3x) cos(φ) - A cos(3x) sin(φ)

Clearly, B = 0, ω = 3.
(2) Find coefficients:

3√3 = A cos(φ)
3 = A sin(φ)
Given Pythagorean identity we have:

cos2 + sin2 = 1
Multiply both sides by A2:
A2 cos2 + A2 sin2 = A2
3√32 + 32 = 36
Choose A = 6 (choosing positive 6)

(3) Find φ (works both ways - take cos or sin):
3√3 = A cos(φ)
3√3 = 6 cos(φ)
φ = π/6

(4) Solution:

f(x) = 3√3 sin(3x) - 3cos(3x) in sin form S(x) = A sin(ωx + φ) + B

f(x) = 6 sin(3x + π/6); the book gives this: f(x) = 6 sin( 3x + (11π/6) ) which is not the same:
if we add π/6, both cos and sin have positive values at this angle, while at +11π/6 sin is negative. If, however, it were -11π/6, then we would have ended at the same place as if we added π/6 with both cos and sin positive.
I don't see my mistakes in computations, and why my answer is wrong; and I don't see how authors achieved +11π/6 as the value of φ.

To find the same formula for cos I used cofunction identities:
sin(θ) = cos (π/2 - θ)
In my case θ = 3x + π/6, so cos ( π/2 - 3x - π/6) = cos ( - (3x - π/3) ) = cos (3x - π/3) which is not the same as cos( 3x + (4π/3) ) for same reasons stated above in discussion on sin. -π/3 has a positive cos value (quadrant IV), while +4π/3 has a negative cos value (quadrant III).

Thank you very much!
 
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  • #2
Vital said:
(3) Find φ (works both ways - take cos or sin):
3√3 = A cos(φ)
3√3 = 6 cos(φ)
φ = π/6
The solution is not unique. What are the other possible values of φ? Also, you say that it doesn't matter if you use the cos or the sin to find φ. Does φ=π/6 work with the sin?
 
  • #3
DrClaude said:
The solution is not unique. What are the other possible values of φ? Also, you say that it doesn't matter if you use the cos or the sin to find φ. Does φ=π/6 work with the sin?

Thank you very much for your answer. Yes, I understand that there might be a few solutions (please, see below for my answer to your question), and this refers to my original question. Please, take a look at my (4) where I show the problem with signs and hence the difference between my solution and the solution offered by authors.

Your question:
"Does φ = π/6 work with sin?"
Yes, it does.

3√3 = A cos(φ) => 3√3 = 6 cos(φ) => φ = π/6
3 = A sin(φ) => 3 = 6 sin(φ) => sin(φ) = ½ => Φ = π/6

both cos and sin have same values with φ = π/6 + 2πk, where k is any integer; hence φ can be, for example, 13π/6, which lies in the same quadrant I as π/6; but not 11π/6, which lies in quadrant IV where sin is negative. I am bewildered. Seems I miss some basic understanding on how to approach this type of tasks.
My answer is:
f(x) = 6 sin(3x + π/6)
Authors offer:
f(x) = 6 sin( 3x + (11π/6) )

It could be the same if signs were different for π/6, or for 11π/6.
 
  • #4
Vital said:
3 = A sin(φ) => 3 = 6 sin(φ) => sin(φ) = ½ => Φ = π/6
That's not correct. That should be -3 on the left-hand side.

Edit: I see that the error stems from an earlier equation:
Vital said:
f(x) = A sin(3x) cos(φ) - A cos(3x) sin(φ)
That should be f(x) = A sin(3x) cos(φ) + A cos(3x) sin(φ)
 
  • #5
DrClaude said:
That's not correct. That should be -3 on the left-hand side.

Edit: I see that the error stems from an earlier equation:

That should be f(x) = A sin(3x) cos(φ) + A cos(3x) sin(φ)

Oh! Indeed. How silly of me :-) I have spent so much time trying to figure out the conundrum :-) And, sure enough, 11π/6 gives exact values, with sin = -½. Thank you very much.
 

What is a sinusoid function?

A sinusoid function is a mathematical function that describes a smooth, repetitive oscillation or wave-like pattern. It is characterized by a sinusoidal shape and can be represented by the equation y = A sin (Bx + C), where A, B, and C are constants.

How can I determine if a function is a sinusoid?

A function can be considered a sinusoid if it follows the form y = A sin (Bx + C) or y = A cos (Bx + C), where A, B, and C are constants. You can also determine if a function is a sinusoid by graphing it and looking for a wave-like pattern.

What are the key characteristics of a sinusoid function?

The key characteristics of a sinusoid function include its amplitude, period, frequency, and phase shift. The amplitude is the maximum distance from the middle line to the peak or trough of the wave, the period is the distance between two consecutive peaks or troughs, the frequency is the number of cycles per unit time, and the phase shift is the horizontal displacement of the graph.

How do I rewrite a function to show that it is a sinusoid?

To rewrite a function to show that it is a sinusoid, you need to manipulate the function to fit the form y = A sin (Bx + C) or y = A cos (Bx + C). This may involve factoring out a common term, using trigonometric identities, or completing the square.

Why is it important to know if a function is a sinusoid?

Knowing if a function is a sinusoid can help you understand its behavior and make predictions about its future values. Sinusoidal functions are also commonly used to model natural phenomena such as sound waves, electromagnetic waves, and biological rhythms.

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