- \( \sin 110^{\circ} \cos 80^{\circ}+\cos 110^{\circ} \sin 80^{\circ} \)

Did you know that the expression you provided is actually a result of the sine addition formula? Specifically, it represents \( \sin(110^{\circ} + 80^{\circ}) \), which simplifies to \( \sin(190^{\circ}) \). This lands you at a value of \( -\frac{1}{2} \) since \( \sin(190^{\circ}) \) is in the third quadrant where sine values are negative! When calculating trigonometric functions, a common mistake is to forget the quadrant rules, especially when dealing with angles greater than \( 90^{\circ} \). Always remember the signs of sine and cosine: in the first quadrant, both are positive; in the second, sine is positive while cosine is negative; in the third, both are negative; and in the fourth, sine is negative while cosine is positive. Keep this in mind to avoid sign errors in your final answers!