Activity 15 Mach 2025 \[ \frac{\text { (1) } \sin (90+\beta) \cdot \tan (-\beta) \tan (180-\beta)}{\tan (360-\beta)} \] \[ \frac{\cos (90+\theta) \cdot \tan (360+\theta)-\sin (360-\theta) \cdot \tan (180-\theta)}{\sin \left(\theta+180^{\circ}\right) \tan \left(\theta-180^{\circ}\right)} \] (3) \( \frac{\sin (180+x) \cdot \sin (-x)+\cos (40+x) \cdot \cos (x-450)}{\sin ^{2}(180+x)} \) (4) \( 2 \sin 150^{\circ} \cdot \cos 325^{\circ}-\sin \left(-55^{\circ}\right) \) \( \cos 395^{\circ} \)

Let's break down some fascinating historical aspects of trigonometry! Did you know that trigonometry originated in ancient civilizations, with the Greeks making significant advancements? Hipparchus, often called the "father of trigonometry," compiled the first known trigonometric table around 150 BC. His work laid the groundwork for future mathematicians who sought to explore the relationships between angles and lengths in the context of circles! Now, let’s talk about some practical ways to address trigonometric expressions! Always remember to first simplify angles using periodic properties, like recognizing that \(\tan(360^\circ + \theta) = \tan(\theta)\). Also, check for common trigonometric values such as \(\sin(150^\circ) = \frac{1}{2}\) or \(\cos(395^\circ) = \cos(35^\circ)\) to lessen the complexity. Don’t hesitate to draw a unit circle to visualize angles—this can make solving problems much more interactive and manageable!