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(a) Sketch, for \( 0 \leq x \leq 2\pi \), the graph of \( y = \sin\left(x + \frac{\pi}{6}\right) \) - Edexcel - A-Level Maths Pure - Question 2 - 2007 - Paper 2
Question 2
(a) Sketch, for \( 0 \leq x \leq 2\pi \), the graph of \( y = \sin\left(x + \frac{\pi}{6}\right) \). (b) Write down the exact coordinates of the points where the gr... show full transcript
Worked Solution & Example Answer:(a) Sketch, for \( 0 \leq x \leq 2\pi \), the graph of \( y = \sin\left(x + \frac{\pi}{6}\right) \) - Edexcel - A-Level Maths Pure - Question 2 - 2007 - Paper 2
Step 1
Sketch, for \( 0 \leq x \leq 2\pi \), the graph of \( y = \sin\left(x + \frac{\pi}{6}\right) \)
Answer
To graph ( y = \sin\left(x + \frac{\pi}{6}\right) ), we should first understand how the transformation affects the sine function. The sine function has a standard period of ( 2\pi ) and swings between (-1) and (1).
- The phase shift of ( \frac{\pi}{6} ) moves the graph to the left by ( \frac{\pi}{6} ).
- Start plotting at ( x = 0 ), which gives us ( \sin\left(0 + \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = 0.5 ).
- The maximum occurs at ( x = \frac{\pi}{2} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} ), where ( y = 1 ) and returns to zero afterward.
- Continue plotting the turning points till ( 2\pi ). Ensure your sketch shows a sine wave with at least two turning points.
Step 2
Write down the exact coordinates of the points where the graph meets the coordinate axes.
Answer
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Intersection with the x-axis:
- Set ( y = 0 ): [\sin\left(x + \frac{\pi}{6}\right) = 0 \Rightarrow x + \frac{\pi}{6} = n\pi \Rightarrow x = n\pi - \frac{\pi}{6}, n \in \mathbb{Z}]
- For ( n = 0 ): ( x = -\frac{\pi}{6} ) (ignore since out of range)
- For ( n = 1 ): ( x = \pi - \frac{\pi}{6} = \frac{5\pi}{6} )
- For ( n = 2 ): ( x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} )
So, intersections: ( \left( \frac{5\pi}{6}, 0 \right) ) and ( \left( \frac{11\pi}{6}, 0 \right) ).
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Intersection with the y-axis:
- When ( x = 0 ): ( y = \sin\left(0 + \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = 0.5 )
- Coordinates: ( \left( 0, 0.5 \right) )
Step 3
Solve, for \( 0 \leq x \leq 2\pi \), the equation \( \sin\left(x + \frac{\pi}{6}\right) = 0.65 \)
Answer
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Solve for x:
- Set: [ x + \frac{\pi}{6} = \arcsin(0.65) + 2k\pi \quad (k \in \mathbb{Z}) \text{ or } x + \frac{\pi}{6} = \pi - \arcsin(0.65) + 2k\pi ]
- Calculate: ( \arcsin(0.65) \approx 0.707 \text{ radians} )
- Set forward:
- Equation 1: [ x + \frac{\pi}{6} = 0.707 \Rightarrow x = 0.707 - \frac{\pi}{6} \approx 0.707 - 0.524 = 0.183 ]
- Equation 2: [ x + \frac{\pi}{6} = \pi - 0.707 \Rightarrow x = \pi - 0.707 - \frac{\pi}{6} \approx 3.141 - 0.707 - 0.524 = 1.910 ]
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Final answers in radians, round to 2 decimal places:
[ x \approx 0.18 ext{ and } 1.91 ]