Understanding the Trigonometric Problem

We are given two equations involving the angles \(x\) and \(y\). Our goal is to use these equations to find the value of the expression \(\sin 3x + \sin 3y\).

• The first equation is: \(\sin x + \sin y = \sqrt{3} (\cos y - \cos x)\).
• The second equation is: \(x + y = \frac{\pi}{2}\).

The condition \(x + y = \frac{\pi}{2}\) suggests that \(x\) and \(y\) are complementary angles. This is a key piece of information we can use to simplify the first equation.

Using the Complementary Angle Relationship

From the second equation, \(x + y = \frac{\pi}{2}\), we can write \(y\) in terms of \(x\):

\(y = \frac{\pi}{2} - x\)

Now we can use the complementary angle identities to relate the trigonometric functions of \(y\) to those of \(x\):

• \(\sin y = \sin \left(\frac{\pi}{2} - x\right) = \cos x\)
• \(\cos y = \cos \left(\frac{\pi}{2} - x\right) = \sin x\)

Simplifying the First Equation Using Identities

Let's substitute the expressions for \(\sin y\) and \(\cos y\) from the complementary angle relationship into the first equation \(\sin x + \sin y = \sqrt{3} (\cos y - \cos x)\):

\(\sin x + (\cos x) = \sqrt{3} ((\sin x) - \cos x)\)

\(\sin x + \cos x = \sqrt{3} \sin x - \sqrt{3} \cos x\)

Solving for the Value of tan x

Now, let's rearrange the terms in the equation to group the \(\sin x\) terms together and the \(\cos x\) terms together:

\(\cos x + \sqrt{3} \cos x = \sqrt{3} \sin x - \sin x\)

Factor out \(\cos x\) on the left side and \(\sin x\) on the right side:

\((1 + \sqrt{3}) \cos x = (\sqrt{3} - 1) \sin x\)

To find \(\tan x\), which is \(\frac{\sin x}{\cos x}\), we can divide both sides by \((\sqrt{3} - 1) \cos x\):

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

\(\tan x = \frac{1 + \sqrt{3}}{\sqrt{3} - 1}\)

To simplify this expression, we can rationalize the denominator by multiplying the numerator and denominator by \((\sqrt{3} + 1)\):

We know that \(\tan \left(\frac{5\pi}{12}\right) = \tan(75^\circ) = 2 + \sqrt{3}\). Given the context of the problem where \(x + y = \frac{\pi}{2}\), we can infer that \(x\) and \(y\) are likely acute angles. Thus, we can identify \(x\) as \(\frac{5\pi}{12}\).

Determining the Values of x and y

We found \(x = \frac{5\pi}{12}\). Using the second equation \(x + y = \frac{\pi}{2}\), we can find \(y\):

\(y = \frac{\pi}{2} - x = \frac{\pi}{2} - \frac{5\pi}{12}\)

To subtract these fractions, find a common denominator, which is 12:

\(y = \frac{6\pi}{12} - \frac{5\pi}{12} = \frac{6\pi - 5\pi}{12} = \frac{\pi}{12}\)

So, \(x = \frac{5\pi}{12}\) and \(y = \frac{\pi}{12}\).

Calculating sin 3x + sin 3y

Now we need to evaluate the expression \(\sin 3x + \sin 3y\) using the values of \(x\) and \(y\) we found.

First, calculate \(3x\) and \(3y\):

• \(3x = 3 \times \frac{5\pi}{12} = \frac{15\pi}{12} = \frac{5\pi}{4}\)
• \(3y = 3 \times \frac{\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4}\)

Now substitute these values into the expression:

\(\sin 3x + \sin 3y = \sin \left(\frac{5\pi}{4}\right) + \sin \left(\frac{\pi}{4}\right)\)

Let's find the values of the individual sine terms:

• \(\sin \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\)
• For \(\sin \left(\frac{5\pi}{4}\right)\), note that \(\frac{5\pi}{4} = \pi + \frac{\pi}{4}\). This angle is in the third quadrant, where the sine function is negative. Using the identity \(\sin(\pi + \theta) = -\sin \theta\):
  \(\sin \left(\frac{5\pi}{4}\right) = \sin \left(\pi + \frac{\pi}{4}\right) = -\sin \left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}\)

Now, add the two values:

\(\sin 3x + \sin 3y = -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = 0\)

Final Answer

The value of \(\sin 3x + \sin 3y\) is \(0\).