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Figure 2 shows a sketch of the curve C with parametric equations $x = 4 ext{sin}igg(t + rac{ ext{π}}{6}igg)$, y = 3 ext{cos}2t, ext{ for } 0 ext{ } < t < 2 ext{π} - Edexcel - A-Level Maths Pure - Question 7 - 2012 - Paper 8
Question 7
Figure 2 shows a sketch of the curve C with parametric equations $x = 4 ext{sin}igg(t + rac{ ext{π}}{6}igg)$, y = 3 ext{cos}2t, ext{ for } 0 ext{ } < t < 2 ex... show full transcript
Worked Solution & Example Answer:Figure 2 shows a sketch of the curve C with parametric equations $x = 4 ext{sin}igg(t + rac{ ext{π}}{6}igg)$, y = 3 ext{cos}2t, ext{ for } 0 ext{ } < t < 2 ext{π} - Edexcel - A-Level Maths Pure - Question 7 - 2012 - Paper 8
Step 1
Find an expression for \( \frac{dy}{dx} \) in terms of t.
Answer
To find ( \frac{dy}{dx} ), we first need to find ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ).
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Compute ( \frac{dx}{dt} ):
[\frac{dx}{dt} = \frac{d}{dt}[4\text{sin}\bigg(t + \frac{\pi}{6}\bigg)] = 4\text{cos}\bigg(t + \frac{\pi}{6}\bigg)]
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Compute ( \frac{dy}{dt} ):
[\frac{dy}{dt} = \frac{d}{dt}[3\text{cos}(2t)] = -6\text{sin}(2t)]
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Now, we can find ( \frac{dy}{dx} ):
[\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-6\text{sin}(2t)}{4\text{cos}\bigg(t +\frac{\pi}{6}\bigg)} = -\frac{3\text{sin}(2t)}{2\text{cos}\bigg(t +\frac{\pi}{6}\bigg)}]
Step 2
Find the coordinates of all the points on C where \( \frac{dy}{dx} = 0 \).
Answer
To find the points where ( \frac{dy}{dx} = 0 ), we set the numerator of ( \frac{dy}{dx} ) to zero:
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Set ( -6\text{sin}(2t) = 0 ):
This gives ( \text{sin}(2t) = 0 ), hence:
[2t = n\pi, \text{ where } n \text{ is an integer}]
Therefore, [t = \frac{n\pi}{2}]
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The values of t in the interval ( 0 < t < 2\pi ) are ( t = \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi ).
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Now, compute the corresponding coordinates for each value:
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For ( t = \frac{\pi}{2} ): [ x = 4\text{sin}\left(\frac{\pi}{2} + \frac{\pi}{6}\right) = 4\text{sin}\left(\frac{2\pi}{3}\right) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} ] [ y = 3\text{cos}(\pi) = -3 \Rightarrow (x, y) = (2\sqrt{3}, -3)]
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For ( t = \pi ): [ x = 4\text{sin}\left(\pi + \frac{\pi}{6}\right) = 4\text{sin}\left(\frac{7\pi}{6}\right) = -2]\ [ y = 3\text{cos}(2\pi) = 3 \Rightarrow (x, y) = (-2, 3)]
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For ( t = \frac{3\pi}{2} ): [ x = 4\text{sin}\left(\frac{3\pi}{2} + \frac{\pi}{6}\right) = -2\sqrt{3}]\ [ y = 3\text{cos}(3\pi) = -3 \Rightarrow (x, y) = (-2\sqrt{3}, -3)]
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For ( t = 2\pi ): [ x = 4\text{sin}\left(2\pi + \frac{\pi}{6}\right) = 2\sqrt{3}]\ [ y = 3\text{cos}(4\pi) = 3 \Rightarrow (x, y) = (2\sqrt{3}, 3)]
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Therefore, the coordinates of all the points on C where ( \frac{dy}{dx} = 0 ) are:
- ( (2\sqrt{3}, -3) )
- ( (-2, 3) )
- ( (-2\sqrt{3}, -3) )
- ( (2\sqrt{3}, 3) )