If anyone would help it would be 4U please help with working out (1 Viewer)

kpad5991 · Feb 27, 2018

altSwift · Feb 27, 2018

kpad5991 said:
https://imgur.com/gallery/7EPaD

$\noindent Please keep in mind I have never derived trig functions before as I'm not up to that chapter, so I just used symbolab.$

$\noindent 14. a)$ \angle QOM = \frac{\pi}{2} $ (A line from the centre of a circle that bisects a chord is perpendicular to the chord)$

$\noindent It is clear that $ \Delta QOM $ is a right-angled triangle and thus $ sin\theta = QM/1 = QM $ (radius is 1)$

$$\noindent Similarly $ cos\theta = OM$

$$\noindent b) $ OP = 1 $ (radius), therefore from a) $ PM = cos\theta + 1$

$$\noindent From a), $ QR = 2sin\theta$

$A = \frac{1}{2}2sin\theta(cos\theta + 1) = sin\theta(cos\theta + 1)$

$\noindent c) $\frac{dA}{d\theta} = cos\theta(cos\theta + 1) - sin^{2}\theta $ (again I have no idea how this is done I just plugged into symbolab, I'm guessing you know how to because you are studying this chapter)$

$$Stationary points occur when $ \frac{dA}{d\theta} = 0$

$cos\theta(cos\theta + 1) - sin^{2}\theta = 0$

$cos\theta(cos\theta + 1) - (1 - cos^{2}\theta) = 0$

$2cos^2\theta + cos\theta - 1 = 0$

$$Solving this quadratic we get $ cos\theta = \frac{1}{2} $ and $ cos\theta = -1$

$Obviously $ cos\theta \neq -1 $ as $ cos\theta $ is a length! (OM)$

$$\noindent So $ cos\theta = \frac{1}{2} $ or $ \theta = \frac{\pi}{3} $ is the only stationary point which we should consider (It's not 420 or 780 or any other multiple of $ \pi $ as $ \theta $ is acute as stated). Using a table of values we can determine that there is a maximum turning point at $ \theta = \frac{\pi}{3} $, and thus the area is maximum when $ \theta = \frac{\pi}{3}$

$\noindent Now $ \Delta PQR = \theta $ (angle at centre is twice angle at circumference), therefore the area of $ \Delta PQR $ is max when $ \theta = \frac{\pi}{3} $, and since $ \Delta PQR $ is isosceles, $ \angle PQR = \angle PRQ = \frac{\pi}{3} $ (which is an equilateral triangle!)$

If anyone would help it would be 4U please help with working out (1 Viewer)

kpad5991

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altSwift

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