Trigonometry - R-Formula and Trigonometric Extrema
Trigonometry - R-Formula and Trigonometric Extrema
The diagram shows a quadrilateral divided into two right-angled triangles and a rectangle.
Given that \(\angle ADC = \theta{}^\circ\), \(AB = 7\) cm, \(AD = 15\) cm and \(\angle BAD = \angle BCD = \angle AFB = 90{}^\circ\).
- Find the length of \(BC\) in the form of \(a \sin \theta - b \cos \theta\). Hence show that the perimeter, \(P\) cm, of the quadrilateral \(ABCD\) is given by \(P = 22 + 22 \sin \theta + 8 \cos \theta\).
- Expressing \(P\) in the form \(P = 22 + R \sin(\theta + \alpha)\).
- Hence state the maximum value of the perimeter and state the corresponding value of \(\theta\).
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Answer:(a) \(BC=15\sin\theta-7\cos\theta\), so \(P=22+22\sin\theta+8\cos\theta\) (b) \(P=22+2\sqrt{137}\sin(\theta+19.983^\circ)\) (c) \(P_{\max}=22+2\sqrt{137}\text{ cm}=45.4\text{ cm}\) when \(\theta=70.0^\circ\)
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