Need help solving for x!

**jsel21** · March 1st 2010, 11:32 AM

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

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

**e^(i*pi)** · March 1st 2010, 11:40 AM

Originally Posted by jsel21

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

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

It seems painfully obvious that there is no value of x for which these two values would be equal. It is like asking when does 2 equal 2+1

**jsel21** · March 1st 2010, 03:34 PM

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.

**jsel21** · January 26th 2011, 11:07 AM

Originally Posted by e^(i*pi)

It seems painfully obvious that there is no value of x for which these two values would be equal. It is like asking when does 2 equal 2+1

I know I posted this a while ago but I just wanted to point out that there are definitely solutions to this. x = pi/3, 4pi/3. I know how to solve in my head but not on paper.

**TheEmptySet** · January 26th 2011, 11:14 AM

Use the identity that

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

This gives the equation

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

Or

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

This will give you the solutions you want.

**LoblawsLawBlog** · January 26th 2011, 05:20 PM

If $x$ is a solution then $\sin(x)$ and $\sin(x+\pi/3)$ 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 $\pi/3$ . 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 $\theta$ .

This leads to the equation $2\theta+\pi/3=\pi$ , since together the three angles make a straight line. This gives you one solution and then the other is easily found.

**Archie Meade** · January 30th 2011, 06:51 AM

Originally Posted by jsel21

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

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

Visually, you could use the Unit Circle.

$sin(x)$ gives the y co-ordinate of a point on the Unit Circle,
hence the y-axis from $-1$ to $1$ acts as an axis of symmetry.

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

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

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

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

to obtain a 2nd solution.

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

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