\( 7.1 \frac{2 \cos \left(90^{\circ}+x\right)-\sin \left(360^{\circ}-x\right)}{3 \sin \left(90^{\circ}-x\right)-\cos \left(180^{\circ}-x\right)} {} 7.2 \frac{\sin \left(180^{\circ}+x\right)}{\tan \left(180^{\circ}-x\right) \tan 45^{\circ} \cdot \cos \left(360^{\circ}-x\right)} {} 7.3 \frac{2 \cos \left(-180^{\circ}-x\right) \cdot \sin (-x)}{\sin \left(x-180^{\circ}\right) \cdot \cos \left(x-360^{\circ}\right)} {} 7.4 \frac{\cos ^{2}\left(180^{\circ}+x\right)}{\sin \left(180^{\circ}-x\right) \cdot \cos \left(x-90^{\circ}\right)}+1 {} 7.5\left[\sin \left(90^{\circ}+x\right) \sin (-x) \tan \left(180^{\circ}-x\right)\right]-\left[\sin \left(90^{\circ}-x\right) \cos \left(x-180^{\circ}\right)\right] \)
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The Deep Dive
Did you know that the angles in trigonometry can reveal intriguing historical connections? The \(90^{\circ}\) and \(180^{\circ}\) angles played significant roles in ancient mathematical studies, especially within Greek geometry. They formed the basis for defining the sine and cosine functions and were essential in shaping the classical world’s understanding of circles and triangles. This laid the groundwork for many modern mathematical principles, showing how our ancestors' ideas continue to influence us today! When working through trigonometric expressions like these, be cautious about common mistakes! A frequently overlooked point is the signs of sine and cosine when dealing with different quadrants of the unit circle. Remember: sine is positive in the first and second quadrants, while cosine is only positive in the first quadrant. Keeping these angles and their respective signs straight is crucial to solving problems accurately—after all, you wouldn't want a small sign mistake to lead to an incorrect conclusion!
