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 \( \sin 2 A \).
Upstudy AI Solution
Answer
Solution

Sign in to Unlock Answers for Free!
A Learning Platform Trusted by Millions of Real Students and Teachers.

Answered by UpStudy AI and reviewed by a Professional Tutor
Mind Expander
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.