\( \frac { \sin 800 \cdot \cos ( - 395 ) - \cos ( 2 \omega ) \cdot \sin ( 235 ) } { \tan 150 ^ { \circ } \cdot \sin 775 + \cos ( - 30 ) \cdot \cos 125 ^ { \circ } } \)

To simplify the expression, we need to recall some trigonometric identities and values. We know that \( \tan 150^\circ = -\frac{1}{\sqrt{3}} \) and that \( \cos(-\theta) = \cos(\theta) \) and \( \sin(-\theta) = -\sin(\theta) \). Thus, we can evaluate each function based on standard angles. Next, let's analyze the actual angles: - \( \sin 800 \) and \( \cos(-395) \) may be simplified using periodic properties. For instance, \( 800^\circ \mod 360 = 80^\circ \) and \( -395^\circ \mod 360 = 325^\circ \). Hence, we can substitute those values into our expression. - By doing the same conversion for the angles involving \( \sin \) and \( \cos \), you can replace these with equivalent angles less than 360 degrees to facilitate calculation. So remember, for a neat trick next time: always convert your angles into primary ones (0 to 360 degrees) before jumping into calculations! For a deeper dive into trigonometric identities and calculus, grab a classic like "Trigonometry For Dummies" or any basic differential calculus textbook. They're great resources that break these concepts down into bite-sized pieces—perfect for anyone wanting to sharpen their math skills! Plus, you can always impress friends with your newfound "trig" knowledge at parties! 🎉