3) Determine an exact value for the expression: sin(5pi )/(4)-cos(11pi )/(6)cot(pi )/(3)

To solve the expression \(\sin \frac{5 \pi}{4} - \cos \frac{11 \pi}{6} \cot \frac{\pi}{3}\), we evaluate each trigonometric function separately.<br /><br />1. **Evaluate \(\sin \frac{5\pi}{4}\):**<br /><br /> The angle \(\frac{5\pi}{4}\) is in the third quadrant where sine is negative. It is equivalent to \(\pi + \frac{\pi}{4}\).<br /><br /> \[<br /> \sin \frac{5\pi}{4} = -\sin \frac{\pi}{4} = -\frac{\sqrt{2}}{2}<br /> \]<br /><br />2. **Evaluate \(\cos \frac{11\pi}{6}\):**<br /><br /> The angle \(\frac{11\pi}{6}\) is in the fourth quadrant where cosine is positive. It is equivalent to \(2\pi - \frac{\pi}{6}\).<br /><br /> \[<br /> \cos \frac{11\pi}{6} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}<br /> \]<br /><br />3. **Evaluate \(\cot \frac{\pi}{3}\):**<br /><br /> \(\cot \theta = \frac{1}{\tan \theta}\). For \(\theta = \frac{\pi}{3}\):<br /><br /> \[<br /> \tan \frac{\pi}{3} = \sqrt{3} \quad \Rightarrow \quad \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}}<br /> \]<br /><br />4. **Combine the results:**<br /><br /> Substitute the evaluated values into the original expression:<br /><br /> \[<br /> \sin \frac{5\pi}{4} - \cos \frac{11\pi}{6} \cot \frac{\pi}{3} = -\frac{\sqrt{2}}{2} - \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}}\right)<br /> \]<br /><br /> Simplify the second term:<br /><br /> \[<br /> \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}} = \frac{1}{2}<br /> \]<br /><br /> Therefore, the expression becomes:<br /><br /> \[<br /> -\frac{\sqrt{2}}{2} - \frac{1}{2}<br /> \]<br /><br />Thus, the exact value of the expression is:<br /><br />\[<br />-\frac{\sqrt{2}}{2} - \frac{1}{2}<br />\]