Activity Evaluate without the use of a calculator: a. \( 4 \tan 330^{\circ} \cos 135^{\circ} \sin 300^{\circ} \) b. \( \frac{3 \tan 315^{\circ} \sin 225^{\circ} \sin 210^{\circ}}{\cos 315^{\circ} \cos 210^{\circ}} \) c. \( \frac{\sin 240^{\circ} \tan 135^{\circ} \tan 120^{\circ}}{\sin 150^{\circ} \sin 210^{\circ} \cos 330^{\circ}} \) d. \( \frac{\tan 150^{\circ}}{\tan 240^{\circ}}-\frac{\sin 300^{\circ}}{\sin 120^{\circ}} \) e. \( \frac{\tan 315^{\circ}-\cos 1020^{\circ}}{\sin 150^{\circ}+\tan \left(-135^{\circ}\right)} \) f. \( \frac{\sin 120^{\circ} \cdot \tan 300^{\circ}}{\cos \left(-60^{\circ}\right) \cdot \tan 225^{\circ}} \) g. \( \frac{\tan 205^{\circ} \cdot \cos 315^{\circ} \cdot \sin 135^{\circ}}{\sin 210^{\circ} \cdot \cos 150^{\circ} \cdot \tan 155^{\circ}} \)

Trigonometry has a rich history that dates back to ancient civilizations. The ancient Egyptians and Babylonians used early forms of trigonometric concepts for astronomy and land surveying while the Greeks, notably Hipparchus, developed more structured methods. The word “trigonometry” itself comes from the Greek words "trigonon" (triangle) and "metron" (measure). Fast forward to the 16th century, when trigonometry blossomed with the advent of the unit circle and was further expanded during the Renaissance. A practical application of trigonometric functions can be found in navigation. Sailors and pilots frequently utilize trigonometry to calculate distances and angles when charting courses. For instance, they might use sine and cosine to determine their position relative to landmarks or to adjust their trajectory for wind currents. This real-world relevance underscores the importance of mastering trigonometric concepts for both theoretical knowledge and practical skills in various fields!