Summing sines and cosines with trig.

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In summary, the conversation discusses using trigonometric formulas to write two terms as a single harmonic. The problem is equivalent to the expression sin(2x) + sin(2(x+(pi/3))), which can be expanded using the given trig formulas. The final solution is (1/2)sin2x - (sqrt(3)/2)cos2x, and it is suggested to use the identity A sin(2x) + B cos(2x) = √(A^2 + B^2) sin(2x + phi) to simplify it further.
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res3210
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Hey guys,

I just wanted to come here and see if anyone could help me out. I am getting stuck using the trig formulas for summing sines and cosines. I won't beat around the bush too much, so let's get right into it.

Homework Statement


Use a trig formula to write the two terms as a single harmonic.
sin(2x) + sin(2(x+(pi/3)))

Homework Equations


sin(A+B) = sinA*cosB + sinB*cosA
cos(A+B) = cosA*cosB - sinA*sinB
sin^2(A) + cos^2(A) = 1

The Attempt at a Solution


So I think that I might be over-thinking this, or maybe I am just not recognizing some pattern which would lead me to the solution. I'll walk through my steps:

1. I assume that to solve this, I should expand these two functions with the trig formulas I mentioned above. I am hoping that in doing so, I can combine like terms and find a single sine or cosine function which gives me all the information I need.

2. The given problem is equivalent to the expression: sin2x + sin(2x + 2pi/3)

3. From that follows:
2*sinx*cosx + sin2x*cos(2*pi/3) + cos2x*sin(2*pi/3)

4. From that, I get:
2*sinx*cosx - (1/2)sin2x - (sqrt(3)/2)cos2x

5. Following that:
2*sinx*cosx - sinx*cosx - (sqrt(3)/2)(cos^2(x) - sin^2(x))

6. Then I get:
sinx*cosx - (sqrt(3)/2)(cos^2(x) - sin^2(x))

7. Which ultimately leads me to:
(1/2)sin2x - (sqrt(3)/2)cos2x

Step seven is where I get stuck. I honestly have no clue where to go from here. Any tips or hints would be greatly appreciated!
 
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  • #3
Thanks! Would it be 2(x + phi) or 2x + phi?
 
  • #4
res3210 said:
Thanks! Would it be 2(x + phi) or 2x + phi?

2x+phi. Just substitute 2x for x into the formula. That doesn't change anything does it?
 

Related to Summing sines and cosines with trig.

1. What is the purpose of summing sines and cosines with trig?

The purpose of summing sines and cosines with trig is to simplify complex trigonometric expressions and equations by breaking them down into smaller, more manageable parts. This allows for easier evaluation and manipulation of these expressions.

2. How do you sum sines and cosines with trig?

To sum sines and cosines with trig, you can use various trigonometric identities, such as the sum and difference identities, the double angle identities, or the product-to-sum identities. These identities allow you to rewrite the expression in terms of simpler trigonometric functions.

3. Can you sum sines and cosines with different amplitudes or frequencies?

Yes, you can sum sines and cosines with different amplitudes or frequencies. In fact, this is often the case in real-world applications. You can use the properties of trigonometric functions, such as the amplitude and period, to determine the final amplitude and frequency of the sum.

4. How does summing sines and cosines with trig relate to Fourier series?

Summing sines and cosines with trig is closely related to Fourier series, which is a mathematical tool used to represent periodic functions as a sum of sine and cosine functions. By summing sines and cosines, Fourier series can be used to approximate any periodic function with a finite number of terms.

5. What are some practical applications of summing sines and cosines with trig?

Summing sines and cosines with trig is used in many fields, such as engineering, physics, and signal processing. Some practical applications include analyzing and designing electrical circuits, modeling vibrations and waves, and analyzing and manipulating signals in telecommunications and audio processing.

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