(4) \( \frac{\sin \left(360^{\circ}-\theta\right)+\cos \left(90^{\circ}+\theta\right)}{\sin (-\theta)} \) (6) \( \frac{\cos \left(\theta-90^{\circ}\right) \cdot \cos (-\theta) \cdot \sin \left(360^{\circ}-\theta\right)}{\sin \left(180^{\circ}+\theta\right) \cdot \cos \left(\theta-360^{\circ}\right)} \)

Did you know that the sine and cosine functions have fascinating properties that are not just useful in math but are also deeply rooted in the world around us? The sine of the angle \(360^{\circ} - \theta\) is equal to \(\sin(\theta)\), demonstrating how angles can wrap around in a circle. Similarly, the cosine function's periodic nature means \(\cos(90^{\circ} + \theta) = -\sin(\theta)\), showing the harmony between these functions! When simplifying the expressions, a common mistake is to overlook the signs and angles when applying trigonometric identities. For example, realizing that \(\sin(-\theta) = -\sin(\theta)\) and leveraging identities like \(\cos(\theta - 90^{\circ}) = \sin(\theta)\) can make your calculations smoother. Always take a moment to double-check the quadrant you're in when evaluating these functions—that's where the magic truly happens!