(ia) \( \cos 330^{\circ} \cdot \sin 60^{\circ} \) \( \begin{array}{ll}\tan 315^{\circ}-2 \cos 60^{\circ}+\sin 210^{\circ} & \text { (b) } \frac{\sin 135^{\circ}}{\cos 225^{\circ}} \\ \text { (c) } & \sin 570^{\circ}+\cos 240^{\circ}-\tan 135^{\circ} \\ \text { (c) } \frac{\tan 330^{\circ}}{\sin 330^{\circ}} & \text { (f) } \frac{\sin 410^{\circ}}{\cos 40^{\circ}} \\ \text { (g) } \sin 150^{\circ}-\tan 240^{\circ} \cdot \cos 210^{\circ} & \text { (b) } \tan 120^{\circ} \cdot \cos 210^{\circ}-\sin ^{2} 315^{\circ} \\ \text { (i) } \frac{\tan 150^{\circ}}{\tan 240^{\circ}}-\frac{\sin 300^{\circ}}{\sin 120^{\circ}} & \text { (id) } \frac{\tan 315^{\circ}-\cos 1020^{\circ}}{\sin 150^{\circ}+\tan \left(-135^{\circ}\right)} \\ \text { (k) } \frac{\tan 225^{\circ}-\sin x}{\sin x+\sin 270^{\circ}} & \text { (l) } \cos \left(-315^{\circ}\right) \cdot \sin 315^{\circ}-\frac{\cos 20^{\circ}}{\sin 250^{\circ}}\end{array} \)
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The Deep Dive
Here's a fun nugget of knowledge! Did you know that the angles in trigonometry often correspond to specific coordinates on the unit circle? For example, \( \cos 330^{\circ} \) corresponds to the coordinate (√3/2, -1/2), while \( \sin 60^{\circ} \) is (1/2, √3/2). When you multiply these values, you're essentially finding a slice of the geometric pizza created by the unit circle, giving a vivid visual for these functions! Now, here’s a cool real-world application: trigonometric functions aren’t just for math class; they play crucial roles in various fields like engineering, physics, and even music! For instance, engineers use trigonometry to determine forces on objects, while sound waves can be modeled using sinusoidal functions, allowing musicians to create the perfect harmony. So next time you hear a song or build something, remember—trigonometry is likely playing a hidden yet vital role!
