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

To find the points where the graph meets the axes, we identify:

• For the x-axis ($y = 0$), we set:

  [ \sin\left(x + \frac{\pi}{6}\right) = 0 ]

The solutions are:

• $x = 0$ when $x + \frac{\pi}{6} = n\pi$ (for integers n). This gives:

  • $x = -\frac{\pi}{6} + n\pi$, checking in the range gives $\left(0, 0\right)$ and $\left(\frac{5\pi}{6}, 0\right)$,
  • $\left(\frac{17\pi}{6}, 0\right)$.

• For the y-axis ($x = 0$), we have:

[ y = \sin\left(0 + \frac{\pi}{6}\right) = \frac{1}{2} ]

Thus, the points where the graph meets the coordinate axes are:

• $\left(0, \frac{1}{2}\right)$ and $\left(\frac{5\pi}{6}, 0\right)$, $\left(\frac{17\pi}{6}, 0\right)$.

To solve for $x$:

1. Rewrite the equation as:

[ x + \frac{\pi}{6} = \arcsin(0.65) ]

2. Calculating:\n
   [ \arcsin(0.65) \approx 0.707 \text{ radians} ]

3. Therefore, we have:

[ x + \frac{\pi}{6} = 0.707 ]

4. Solve for $x$:

[ x = 0.707 - \frac{\pi}{6} \approx 0.18 \text{ radians} ]

5. Additionally, since sine is positive in the first and second quadrants:

[ x + \frac{\pi}{6} = \pi - 0.707 \approx 2.43 \text{ radians} ]

6. The values of $x$ are:

• $x_1 \approx 0.18$ and $x_2 \approx 2.43$. Finally, rounding:

• $x \approx 1.91$ and $x \approx 4.54$.