Find an equivalent equation involving trig functions

In summary, Rewrite the given equation, attempt 1: ##2\sin(x)\cos(x) + 2\sin(x) + 2\cos(x) = 0##Attempt 2:##2\sin(x)\cos(x) + 2\sqrt{2}*\sin(x + \pi/4) = 0##Both attempts produce the same result.
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Homework Statement
Given the equation $$\sin(2x)+2\sin(x)+2\cos(x) = 0$$ find an equivalent equation on the form ##A+\sin(kx + v)=0## (i.e find the values of the parameters A, k and v).
Relevant Equations
$$\sin(2x) = 2\sin(x)\cos(x)$$ $$2\sin(x)+2\cos(x) = 2\sqrt{2}\sin(x + \pi/4)$$
Rewrite the given equation, attempt 1:
##2\sin(x)\cos(x) + 2\sin(x) + 2\cos(x) = 0##
##\sin(x)\cos(x) + \sin(x) + \cos(x) = 0##
##\sin(x)(\cos(x) + 1) + \cos(x) = 0##, naaah, can't get any relevant out from here.

Attempt 2:
##2\sin(x)\cos(x) + 2\sqrt{2}*\sin(x + \pi/4) = 0##
##\sin(x)\cos(x) + \sqrt{2}\sin(x + \pi/4) = 0##, nope, don't know how continue.

I know I can rewrite ##\cos(x)## as ##\sin(\pi/2 - x)##, but I don't see how it should help.
 
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  • #2
nickek said:
Homework Statement:: Given the equation sin(2x)+2*sin(x)+2*cos(x), find an equivalent equation on the form A+sin(kx + v)=0 (i.e find the values of the parameters A, k and v).
Relevant Equations:: sin(2x) = 2*sin(x)*cos(x), 2*sin(x)+2*cos(x) = 2*sqrt(2)*sin(x + pi/4).

Rewrite the given equation, attempt 1:
2*sin(x)*cos(x) + 2*sin(x) + 2*cos(x) = 0
sin(x)*cos(x) + sin(x) + cos(x) = 0
sin(x)*(cos(x) + 1) + cos(x) = 0, naaah, can't get any relevant out from here.

Attemt 2:
2*sin(x)*cos(x) + 2*sqrt(2)*sin(x + pi/4) = 0
sin(x)*cos(x) + sqrt(2)*sin(x + pi/4) = 0, nope, don't know how continue.

I know I can rewrite cos(x) as sin(pi/2 - x), but I don't see how it should help.
##\sin(2x) +2\sin(x)+2\cos(x)## isn't an equation. And as a function, it is not of the form ##A+\sin(kx+v)## as you can see here:
https://www.wolframalpha.com/input?i=sin(x)cos(x)+sin(x)+cos(x)+=

Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/
 
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You could try ##\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}## and ##\cos(x)=\dfrac{e^{ix}+e^{-ix}}{2}## to find the zeros.
Or use the Weierstraß substitution ##t=\tan(x/2)## for ##|x|<\pi.##

If we look at the solution
1676731846051.png

then there is an asymmetry which seems as if there will be two different solutions for the triple ##(A,k,v).##
 
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  • #5
nickek said:
Attempt 2: ##2\sin(x)\cos(x) + 2\sqrt{2}*\sin(x + \pi/4) = 0##
This looks like a good start to me, with this change.
##\sin(2x) + 2\sqrt{2}*\sin(x + \pi/4) = 0##

You didn't show how you got from ##2\sin(x) + 2\cos(x)## to ##2\sqrt 2\sin(x + \pi/4)##, but I assume that you know what you did and where the factor of ##2\sqrt 2## came from.

If you can convert ##A\sin(2x) + B\sin(x + \pi/4)## into an expression like ##C\sin(kx + v)## using the same technique that produced ##2\sqrt 2 \sin(x + \pi/4)##, that should do it for you. It's possible that the constant A is zero, but I haven't worked things out.

BTW, it seems to me that you really aren't working with an equation, but rather, rewriting the expression involving sin(2x), sin(x), and cos(x) into an identically equal expression. But then again, I don't know what the exact wording of the problem is.
 
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1. What is an equivalent equation involving trig functions?

An equivalent equation involving trig functions is an equation that has the same solution as the original equation, but is written in a different form using trigonometric functions such as sine, cosine, tangent, etc.

2. Why would I need to find an equivalent equation involving trig functions?

There are several reasons why you may need to find an equivalent equation involving trig functions. One common reason is to simplify a complex equation, making it easier to solve. Another reason is to make a connection between different trigonometric functions and their relationships, which can be useful in solving more complex problems.

3. How do I find an equivalent equation involving trig functions?

To find an equivalent equation involving trig functions, you can use trigonometric identities, which are formulas that relate different trigonometric functions. You can also use algebraic manipulation techniques, such as factoring and expanding, to transform the equation into a different form.

4. Can I use any trigonometric function to create an equivalent equation?

Yes, you can use any trigonometric function to create an equivalent equation. However, it is important to keep in mind that some trigonometric functions may be more useful than others, depending on the specific problem you are trying to solve.

5. Are there any tips for finding an equivalent equation involving trig functions?

One helpful tip is to review and memorize common trigonometric identities, as they can be used to transform equations into equivalent forms. Another tip is to practice algebraic manipulation techniques, which can also be useful in transforming equations. Lastly, it is important to carefully consider the problem and choose the most appropriate trigonometric function to use in creating an equivalent equation.

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