The diagram below shows a planning map of a planting area that contains a walking path, $DA$, \(AB

Question

The diagram below shows a planning map of a planting area that contains a walking path, $DA$, $AB$ and $BC$, where $DA = 34$ m, $AB = 12$ m, $\angle DAB = \angle BCD = 90^\circ$ and the acute angle $\angle ADC = \theta$ can vary. Show that $L$ m, the length of the walking path can be expressed as $L = 46 + 34 \sin \theta - 12 \cos \theta$. Express $L$ in the form $p + R \sin(\theta - \alpha)$, where $p$, $R > 0$ and $0^\circ < \alpha < 90^\circ$. Given that the exact length of the walking path is $62$ m, find the value of $\theta$.

Accepted Answer

(i) $L=46+34\sin\theta-12\cos\theta$ (shown). (ii) $p=46$, $R=10\sqrt{13}$, $\alpha=\tan^{-1}\!\left(\dfrac6{17}\right)\approx19.4^\circ$. (iii) $\theta\approx45.8^\circ$.

Trigonometry - Trigonometric Modelling

Trigonometry - Trigonometric Modelling

Solution:

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