Mathematics Grade 12 Hech zars QUESTION 2 2.1 Simplify the following: (6) \[ \frac{\cos \left(x-360^{\circ}\right) \cdot \sin \left(360^{\circ}-x\right) \cdot \cos \left(90^{\circ}+x\right)}{\sin ^{2}\left(90^{\circ}-x\right) \cdot \sin (-x)} \] 2.2 Given: \( \cos (x-y)=\cos x \cdot \cos y+\sin x \cdot \sin y \) Derive \( \sin (x+y) \) (3) 2.3 Consider: \( \sin \left(2 x+40^{\circ}\right) \cos \left(x+30^{\circ}\right)-\cos \left(2 x+40^{\circ}\right) \sin \left(x+30^{\circ}\right) \) 2.3.1 Write as a single trigonometric form in its simplest form. (2) 2.3.2 Hence determine the general solution of the equation: \[ \sin \left(2 x+40^{\circ}\right) \cos \left(x+30^{\circ}\right)-\cos \left(2 x+40^{\circ}\right) \sin \left(x+30^{\circ}\right)=-\frac{1}{2} \] [15]
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Have you ever noticed how trigonometry is deeply intertwined with history? It actually dates back to ancient civilizations, like the Babylonians and Egyptians, who used it not just for astronomy but also for architecture and navigation. The ancient Greeks, particularly with figures such as Hipparchus and Ptolemy, expanded on these concepts, laying the groundwork for the sine and cosine functions we use today. Trigonometric functions weren't just scholarly pursuits; they had real applications in making sense of the heavens! In practical terms, understanding how to manipulate trigonometric identities can save you a lot of headaches, especially in fields like physics and engineering. For instance, simplifying expressions can help you solve for unknown variables in mechanics or electrical circuits. A common mistake is forgetting to apply angle properties like co-function identities or the periodic nature of sine and cosine. Remember, keeping your work organized and applying properties step by step can help avoid those pesky errors! Keep up the math magic!