Mathematics/ Term 1 Test QUESTION 7 Given the compound angle formula: \( \quad \cos (A-B)=\cos A \cdot \cos B+\sin A . \sin B \) \( 7.1 \quad \) Use the above formula to derive the specific formula for \( \sin 2 A \).

To derive the formula for \( \sin 2A \) using the compound angle formula, we can utilize the fact that \( 2A \) can be expressed as \( A + A \). So, applying the angle addition formula for sine, we get: \[ \sin(2A) = \sin(A + A) = \sin A \cdot \cos A + \cos A \cdot \sin A \] Since both terms are identical, we can simplify it to: \[ \sin(2A) = 2 \sin A \cdot \cos A \] And there you have it! The specific formula for \( \sin 2A \) is \( \sin(2A) = 2 \sin A \cdot \cos A \). To have some fun with it, just think of it like a dance! When \( A \) takes the stage with its sine and cosine moves, double the action makes for one spectacular \( 2A \) performance—it's all in the rhythm of mathematics! Now, just to add a little extra flavor, this nifty formula opens doors to many applications in real-world scenarios, especially in engineering and physics, where calculating angles and periodic functions are essential. Whether you're adjusting the angle of a satellite dish for the best signal or predicting the tides, knowing how to derive and apply \( \sin(2A) \) is a key part of the toolkit.