Trigonometry - R-Formula and Trigonometric Extrema
Trigonometry - R-Formula and Trigonometric Extrema
The diagram below shows the chords \(PQ\) and \(RS\) and they intersect at the point \(X\). Given that \(PQ=7\) cm, \(RS=24\) cm, \(\angle RXQ=90{}^\circ\) and \(\angle PRS=\theta\) where \(\theta\) varies.
- Prove that \(PR = 24 \cos \theta + 7 \sin \theta\).
- Express \(PR\) in the form \(R \cos(\theta - \alpha)\).
- Find the maximum length of \(PR\) and the corresponding value of \(\theta\).
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Answer:(a) \(PR=24\cos\theta+7\sin\theta\) (b) \(PR=25\cos(\theta-16.3^\circ)\) (c) \(PR_{\max}=25\text{ cm}\) when \(\theta=16.3^\circ\)
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