Leaving Cert Mathematics Trigonometry: The diagram below shows the grap

The diagram below shows the graph of the function $f : x \mapsto \sin 2x$ - Leaving Cert Mathematics - Question 5 - 2014

Question 5

$The-diagram-below-shows-the-graph-of-the-function--$f-:-x-\mapsto-\sin-2x$-Leaving Cert Mathematics-Question 5-2014.png$

The diagram below shows the graph of the function $f : x \mapsto \sin 2x$. The line $2y = 1$ is also shown. (a) On the same diagram above, sketch the graphs of $g ... show full transcript

Worked Solution & Example Answer:The diagram below shows the graph of the function $f : x \mapsto \sin 2x$ - Leaving Cert Mathematics - Question 5 - 2014

Step 1

On the same diagram above, sketch the graphs of $g : x \mapsto \sin x$ and $h : x \mapsto 3\sin 2x$. Indicate clearly which is $g$ and which is $h$.

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Answer

To sketch the graphs of the functions:

Graph of $g : x \mapsto \sin x$ :
- This is a standard sine wave oscillating between -1 and 1 with a period of $2\pi$ . It crosses the x-axis at integer multiples of $\pi$ .
Graph of $h : x \mapsto 3\sin 2x$ :
- This function oscillates between -3 and 3 due to the amplitude of 3, and it has a period of $\pi$ because the frequency is doubled (2 in front of x). It also crosses the x-axis at $x = \frac{n\pi}{2}$ where $n$ is any integer.
Indicate the functions:
- Use different colors for clarity: let $g$ be in green and $h$ in red. Clearly label both graphs on the diagram.

Step 2

Find the co-ordinates of the point $P$ in the diagram.

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Answer

To find the coordinates of point $P$ which is the intersection of the line $y = \frac{1}{2}$ and the curve $y = \sin 2x$ :

Setting up the equation:
- Set $\sin 2x = \frac{1}{2}$ .
Solving for $x$ :
- The general solution for $\sin \theta = \frac{1}{2}$ is $\theta = \frac{\pi}{6} + 2n\pi$ or $\theta = \frac{5\pi}{6} + 2n\pi$ , where $n \in \mathbb{Z}$ .
- For $\theta = 2x$ $θ = 2 x$ , we have:
  1. $2x = \frac{\pi}{6} + 2n\pi$ $2 x = \frac{π}{6} + 2 nπ$
    - This gives $x = \frac{\pi}{12} + n\pi$
  2. $2x = \frac{5\pi}{6} + 2n\pi$ $2 x = \frac{5 π}{6} + 2 nπ$
    - This gives $x = \frac{5\pi}{12} + n\pi$
Finding suitable $x$ values:
- Since we are considering $x$ $x$ in the range $\left[0, 2\pi\right]$ $[0, 2 π]$ , the valid values from these solutions are:
  - $x = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{13\pi}{12}, \frac{17\pi}{12}$ .
Identifying the y-coordinate of point $P$ :
- The y-coordinate is clearly $\frac{1}{2}$ since point $P$ lies on the line $y = \frac{1}{2}$ .

Thus, the coordinates of point $P$ are:

o $\left(\frac{17\pi}{12}, \frac{1}{2}\right)$ .

The diagram below shows the graph of the function $f : x \mapsto \sin 2x$ - Leaving Cert Mathematics - Question 5 - 2014

Worked Solution & Example Answer:The diagram below shows the graph of the function $f : x \mapsto \sin 2x$ - Leaving Cert Mathematics - Question 5 - 2014

On the same diagram above, sketch the graphs of $g : x \mapsto \sin x$ and $h : x \mapsto 3\sin 2x$. Indicate clearly which is $g$ and which is $h$.

Find the co-ordinates of the point $P$ in the diagram.

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