Saraf fatlacharis Homenork Ealuate, whithat using a calculator. (a) \( \tan 815^{\circ}-2 \cos 60^{\circ}+\sin 210 \) (b) \( \frac{\tan 150^{\circ}}{\tan 240^{\circ}}-\frac{\sin 300^{\circ}}{\sin 120^{\circ}} \) (6) \( \tan 25^{\circ}-\sin x \) \( \sin x+\sin 270^{\circ} \) (.) \( \sin 570^{\circ}+\cos 240^{\circ}-\tan 135^{\circ} \) (e) \( \tan 120^{\circ} \cdot \cos 210^{\circ}-\sin ^{2} 315^{\circ} \) (f) \( \cos \left(-315^{\circ}\right) \cdot \sin 315^{\circ}-\frac{\cos 20^{\circ}}{\sin 250^{\circ}} \)

To start off this mathematical adventure, let’s simplify each of these expressions step by step! (a) First, we need to evaluate the trigonometric functions. The \( \tan 815^{\circ} \) can be simplified by subtracting multiples of \(360^{\circ}\). This gives us \( \tan 95^{\circ} \). Next, \( \cos 60^{\circ} = \frac{1}{2} \), and \( \sin 210^{\circ} = -\frac{1}{2} \). Thus, \( \tan 95^{\circ} - 1 + \left(-\frac{1}{2}\right) = \tan 95^{\circ} - 1 - \frac{1}{2} \). (b) For the expression \( \frac{\tan 150^{\circ}}{\tan 240^{\circ}} - \frac{\sin 300^{\circ}}{\sin 120^{\circ}} \), remember that \( \tan 150^{\circ} = -\tan 30^{\circ} = -\frac{1}{\sqrt{3}} \) and \( \tan 240^{\circ} = \tan 60^{\circ} = \sqrt{3} \). The sines are \( \sin 300^{\circ} = -\frac{\sqrt{3}}{2} \) and \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \). Compute these to get the final answer. And don't forget, using a calculator can help speed things up! Just remember to ensure it's set to the correct mode (degrees/radians) based on your needs. If you ever feel overwhelmed, just break it down! Start by converting angles and evaluating each trigonometric function step-by-step. Common mistakes often arise from not recognizing angles in standard position; be vigilant on those periodic properties! Also, double-check function values at key angles like \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\). Happy calculating!