Evaluate `(cos3 theta-2cos 4theta)/(sin 3 theta+2sin 4theta),` when `theta=150^(@)`.

Answer

To evaluate the expression \(\frac{\cos 3\theta - 2\cos 4\theta}{\sin 3\theta + 2\sin 4\theta}\) when \(\theta = 150^\circ\), we will follow these steps:

### Step 1: Substitute \(\theta\) into the expression
First, we substitute \(\theta = 150^\circ\) into the expression:

\[ \cos 3\theta = \cos(3 \times 150^\circ) = \cos(450^\circ) \]
\[ \cos 4\theta = \cos(4 \times 150^\circ) = \cos(600^\circ) \]
\[ \sin 3\theta = \sin(3 \times 150^\circ) = \sin(450^\circ) \]
\[ \sin 4\theta = \sin(4 \times 150^\circ) = \sin(600^\circ) \]

### Step 2: Simplify the angles
Now, we simplify the angles using the periodic properties of sine and cosine:

- For \(\cos(450^\circ)\):
\[ \cos(450^\circ) = \cos(360^\circ + 90^\circ) = \cos(90^\circ) = 0 \]

- For \(\cos(600^\circ)\):
\[ \cos(600^\circ) = \cos(720^\circ - 120^\circ) = \cos(-120^\circ) = -\cos(120^\circ) = -\left(-\frac{1}{2}\right) = \frac{1}{2} \]

- For \(\sin(450^\circ)\):
\[ \sin(450^\circ) = \sin(360^\circ + 90^\circ) = \sin(90^\circ) = 1 \]

- For \(\sin(600^\circ)\):
\[ \sin(600^\circ) = \sin(720^\circ - 120^\circ) = -\sin(120^\circ) = -\left(\frac{\sqrt{3}}{2}\right) = -\frac{\sqrt{3}}{2} \]

### Step 3: Substitute simplified values into the expression
Now we substitute these values back into the expression:

\[ \frac{\cos 450^\circ - 2\cos 600^\circ}{\sin 450^\circ + 2\sin 600^\circ} = \frac{0 - 2 \cdot \frac{1}{2}}{1 + 2 \cdot \left(-\frac{\sqrt{3}}{2}\right)} \]

This simplifies to:

\[ \frac{0 - 1}{1 - \sqrt{3}} = \frac{-1}{1 - \sqrt{3}} \]

### Step 4: Rationalize the denominator
To simplify further, we can multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{-1}{1 - \sqrt{3}} \cdot \frac{1 + \sqrt{3}}{1 + \sqrt{3}} = \frac{-1(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{-(1 + \sqrt{3})}{1 - 3} = \frac{-(1 + \sqrt{3})}{-2} = \frac{1 + \sqrt{3}}{2} \]

### Final Answer
Thus, the value of the expression is:

\[ \frac{1 + \sqrt{3}}{2} \]