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9. (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 10 - 2007 - Paper 2
Question 10
9. (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 g... show full transcript
Worked Solution & Example Answer:9. (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 10 - 2007 - Paper 2
Step 1
(a) Sketch the graph
Answer
To sketch the graph of ( y = \sin\left(x + \frac{\pi}{6}\right) ) over the interval ( 0 \leq x \leq 2\pi ):
- Determine the amplitude and period: The amplitude is 1 and the period is ( 2\pi ).
- Phase shift: The graph is shifted to the left by ( \frac{\pi}{6} ), so the starting point is at ( -\frac{\pi}{6} ) (which is outside the interval and thus ignored).
- Key points: Calculate points at which the function intersects the axes and its turning points:
- At ( x = 0 ): ( y = \sin\left(0 + \frac{\pi}{6}\right) = \frac{1}{2} )
- At ( x = \frac{\pi}{6} ): ( y = \sin(0) = 0 )
- At ( x = \frac{5\pi}{6} ): ( y = \sin\left(\frac{5\pi}{6} + \frac{\pi}{6}\right) = \sin\pi = 0 )
- At ( x = \frac{3\pi}{2} ): ( y = \sin\left(\frac{3\pi}{2} + \frac{\pi}{6}\right) = -\frac{1}{2} )
- At ( x = 2\pi ): ( y = \sin\left(2\pi + \frac{\pi}{6}\right) = \frac{1}{2} )
- Sketch the sine wave through the calculated points ensuring it goes above and below the x-axis as appropriate.
Step 2
Step 3
(c) Solve for \( \sin\left(x + \frac{\pi}{6}\right) = 0.65 \)
Answer
To solve the equation ( \sin\left(x + \frac{\pi}{6}\right) = 0.65 ), we proceed as follows:
- Find the reference angle: ( x + \frac{\pi}{6} = \arcsin(0.65) ) This gives approximately ( 0.707 , \text{radians} ).
- Consider both quadrants where sine is positive:
- First solution: ( x + \frac{\pi}{6} = 0.707 ) → ( x = 0.707 - \frac{\pi}{6} \approx 0.18 , \text{radians} )
- Second solution: ( x + \frac{\pi}{6} = \pi - 0.707 \approx 2.434 ) → ( x = 2.434 - \frac{\pi}{6} \approx 2.14 , \text{radians} )
- Check the solutions are within the interval: Both ( 0.18 ) and ( 2.14 ) are valid solutions within the range ( [0, 2\pi] ).