sin(x) = sin(x + pi/3)

I know this is simple but so am i... thanks in advance for any help

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- Mar 1st 2010, 11:32 AMjsel21Need help solving for x!
sin(x) = sin(x + pi/3)

I know this is simple but so am i... thanks in advance for any help - Mar 1st 2010, 11:40 AMe^(i*pi)
- Mar 1st 2010, 03:34 PMjsel21
but there are solutions. When entering both of these in my graping calculator there are multiple points in which the two functions intersect. I just dont know how to solve for them. I'm looking for the the solutions between 0 and 2pi.

- Jan 26th 2011, 11:07 AMjsel21
- Jan 26th 2011, 11:14 AMTheEmptySet
Use the identity that

$\displaystyle \sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$

This gives the equation

$\displaystyle \sin(x)=\frac{1}{2}\sin(x)+\frac{\sqrt{3}}{2}\cos( x)$

Or

$\displaystyle \tan(x)=\sqrt{3}$

This will give you the solutions you want. - Jan 26th 2011, 05:20 PMLoblawsLawBlog
If $\displaystyle x$ is a solution then $\displaystyle \sin(x)$ and $\displaystyle \sin(x+\pi/3)$ must lie at the intersections of some horizontal line through the unit circle. Furthermore, the angle between these points (measured from the origin) must be $\displaystyle \pi/3$. But the other angles between the rightmost point and the positive x axis and the leftmost point and the negative x axis must be equal by symmetry, so call them $\displaystyle \theta$.

This leads to the equation $\displaystyle 2\theta+\pi/3=\pi$, since together the three angles make a straight line. This gives you one solution and then the other is easily found. - Jan 30th 2011, 06:51 AMArchie Meade
Visually, you could use the Unit Circle.

$\displaystyle sin(x)$ gives the y co-ordinate of a point on the Unit Circle,

hence the y-axis from $\displaystyle -1$ to $\displaystyle 1$ acts as an axis of symmetry.

$\displaystyle 0\le\ x\ \le \pi\Rightarrow\ sin(x)=sin\left(x+60^o\right)$

requires that $\displaystyle x$ and $\displaystyle x+60^0$ are $\displaystyle 30^o$ either side of the y-axis.

$\displaystyle x=90^o-30^o=60^o$

The exact same logic may be used underneath the x-axis for $\displaystyle \pi\le\ x\ \le\ 2\pi$

to obtain a 2nd solution.

Complete the analysis by adding multiples of 360 degrees to the two solutions.