If `sintheta+costheta=1/2` then find the value of `16(sin2theta+cos4theta+sin6theta)`

Answer

To solve the problem, we need to find the value of \( 16(\sin 2\theta + \cos 4\theta + \sin 6\theta) \) given that \( \sin \theta + \cos \theta = \frac{1}{2} \).

### Step-by-Step Solution:

1. **Square the Given Equation**:
\[ (\sin \theta + \cos \theta)^2 = \left(\frac{1}{2}\right)^2 \]
Expanding the left side:
\[ \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta = \frac{1}{4} \]

2. **Use the Pythagorean Identity**:
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). Thus, we can substitute:
\[ 1 + 2\sin \theta \cos \theta = \frac{1}{4} \]

3. **Solve for \( \sin 2\theta \)**:
Since \( \sin 2\theta = 2\sin \theta \cos \theta \), we can rearrange the equation:
\[ 2\sin \theta \cos \theta = \frac{1}{4} - 1 \]
\[ 2\sin \theta \cos \theta = -\frac{3}{4} \]
Therefore,
\[ \sin 2\theta = -\frac{3}{4} \]

4. **Find \( \cos 4\theta \)**:
We use the double angle identity for cosine:
\[ \cos 4\theta = 1 - 2\sin^2 2\theta \]
First, we need \( \sin^2 2\theta \):
\[ \sin^2 2\theta = \left(-\frac{3}{4}\right)^2 = \frac{9}{16} \]
Now substituting back:
\[ \cos 4\theta = 1 - 2\left(\frac{9}{16}\right) = 1 - \frac{18}{16} = 1 - \frac{9}{8} = -\frac{1}{8} \]

5. **Find \( \sin 6\theta \)**:
We can use the identity for \( \sin 3\theta \):
\[ \sin 6\theta = \sin(3 \cdot 2\theta) = 3\sin 2\theta - 4\sin^3 2\theta \]
We already have \( \sin 2\theta = -\frac{3}{4} \), so we need \( \sin^3 2\theta \):
\[ \sin^3 2\theta = \left(-\frac{3}{4}\right)^3 = -\frac{27}{64} \]
Now substituting:
\[ \sin 6\theta = 3\left(-\frac{3}{4}\right) - 4\left(-\frac{27}{64}\right) \]
Simplifying:
\[ \sin 6\theta = -\frac{9}{4} + \frac{108}{64} = -\frac{9}{4} + \frac{27}{16} \]
Convert \( -\frac{9}{4} \) to sixteenths:
\[ -\frac{9}{4} = -\frac{36}{16} \]
Thus,
\[ \sin 6\theta = -\frac{36}{16} + \frac{27}{16} = -\frac{9}{16} \]

6. **Combine the Results**:
Now we can substitute back into the expression:
\[ 16(\sin 2\theta + \cos 4\theta + \sin 6\theta) = 16\left(-\frac{3}{4} - \frac{1}{8} - \frac{9}{16}\right) \]

7. **Finding a Common Denominator**:
The common denominator for \( 4, 8, \) and \( 16 \) is \( 16 \):
\[ -\frac{3}{4} = -\frac{12}{16}, \quad -\frac{1}{8} = -\frac{2}{16}, \quad -\frac{9}{16} = -\frac{9}{16} \]
Adding these:
\[ -\frac{12}{16} - \frac{2}{16} - \frac{9}{16} = -\frac{23}{16} \]

8. **Final Calculation**:
Now multiply by \( 16 \):
\[ 16\left(-\frac{23}{16}\right) = -23 \]

### Final Answer:
\[ \boxed{-23} \]