Trigonometry - Solving Trigonometric Equations

In summary, the conversation discusses how to rewrite the expression ##-sin 2x + \sqrt{3}cos 2x## as ##A \sin(2x+\phi)## without using the formula ##a sin (x) + b cos (x) = c sin (x+v), \ \ c=\sqrt{a^2+b^2} \ ,\ \ \tan v = \frac{b}{a}##. The solution is to use the simpler approach of writing the expression as ##2sin(2x + 2\pi/3)##. It is mentioned that it does not matter which value of A is used as long as it corresponds to the correct angle.
  • #1
Rectifier
Gold Member
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The problem

I am trying to write ##-sin 2x + \sqrt{3} \cos 2x ## as ## A \sin(2x+\phi)## without using this formula below ##a sin (x) + b cos (x) = c sin (x+v), \ \ c=\sqrt{a^2+b^2} \ ,\ \ \tan v = \frac{b}{a}##

The attempt

##A \sin(2x+\phi) = A \cos(\phi) \sin(2x) + A \sin(\phi) \cos(2x) ## comparison with the starting expression.

[tex]\begin{cases} A \cos(\phi) = -1 \\ A \sin(\phi) = \sqrt{3} \end{cases}[/tex]

## \phi : ##
[tex]\frac{A \sin(\phi)}{A \cos(\phi)} = \tan \phi = \frac{\sqrt{3}}{-1} = - \sqrt{3} \\ \tan -\phi = \sqrt{3} \\ \phi = -\frac{\pi}{3} + \pi n, \ \ n \in \mathbb{Z}[/tex]

For ## n=0 ## -> ##-\frac{\pi}{3}## gives $$A \sin(-\frac{\pi}{3}) = \sqrt{3} \\ -A \sin(\frac{\pi}{3}) = \sqrt{3} \\ -A \frac{\sqrt{3}}{2}=\sqrt{3} \\ A = -2$$

##n = 1## from above gives ## \phi = \frac{2 \pi}{3} ##
$$ A \sin(\phi) = \sqrt{3} \\ A \sin(\frac{2 \pi}{3}) = \sqrt{3}$$

I can rewrite ## \sin(\frac{2 \pi}{3}) ## as ## 2 \sin(\frac{\pi}{3}) \cos(\frac{\pi}{3}) = 2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2}##

$$ A \sin(\frac{2 \pi}{3}) = \sqrt{3} \\ A \frac{\sqrt{3}}{2} = \sqrt{3} \\ A = 2$$

Which A should I use in my answer? And why?
 
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  • #2
An easier approach
$$-sin 2x + \sqrt{3}cos 2x = 2\left ( -\frac{1}{2}sin 2x+\frac{\sqrt{3}}{2} cos 2x\right )=2\left ( cos (2\pi/3) sin 2x+sin(2\pi/3)cos 2x\right )=2sin(2x + 2 \pi/3)$$
 
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Likes Chestermiller and Rectifier
  • #3
Thank you for that elegant solution. I solved the problem myself just now, it does not really matter which A I use as long as I use it with the corresponding angle, since ## -2 \sin(2x- \frac{\pi}{3}) = 2 \sin(2x + \frac{2 \pi}{3}) ## .
 

What is a sinusoid in trigonometry?

A sinusoid in trigonometry is a mathematical function that represents a periodic oscillation, such as sound or light waves. It is often described as a sine or cosine wave and is used to model various natural phenomena.

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