Figure 1 shows the curve $C$, with equation $y = 6 \, ext{cos} \, x + 2.5 \, ext{sin} \, x$ for $0 \leq x \leq 2\pi$. (a) Express $6 \text{cos} \, x + 2.5 \text{sin} \, x$ in the form $R \text{cos} (x - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$. Give your value of $R$ to 3 decimal places. (b) Find the coordinates of the points on the graph where the curve $C$ crosses the coordinate axes. A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later. She models her results with the continuous function given by $H = 12 + 6 \text{cos} \left( \frac{2\pi t}{52} \right) + 2.5 \text{sin} \left( \frac{2\pi t}{52} \right)$ where $H$ is the number of hours of daylight and $t$ is the number of weeks since her first recording. Use this function to find (c) the maximum and minimum values of $H$ predicted by the model, (d) the values of $t$ when $H = 16$, giving your answers to the nearest whole number. You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.

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Figure 1 shows the curve $C$, with equation $y = 6 \, ext{cos} \, x + 2.5 \, ext{sin} \, x$ for $0 \leq x \leq 2\pi$ - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 6

Question 2

$Figure-1-shows-the-curve-$C$,-with-equation-$y-=-6-\,--ext{cos}-\,-x-+-2.5-\,--ext{sin}-\,-x$-for-$0-\leq-x-\leq-2\pi$-Edexcel-A-Level Maths Pure-Question 2-2014-Paper 6.png$

Figure 1 shows the curve $C$, with equation $y = 6 \, ext{cos} \, x + 2.5 \, ext{sin} \, x$ for $0 \leq x \leq 2\pi$. (a) Express $6 \text{cos} \, x + 2.5 \text{s... show full transcript

Worked Solution & Example Answer:Figure 1 shows the curve $C$, with equation $y = 6 \, ext{cos} \, x + 2.5 \, ext{sin} \, x$ for $0 \leq x \leq 2\pi$ - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 6

Step 1

Express $6 \text{cos} \, x + 2.5 \text{sin} \, x$ in the form $R \text{cos} (x - \alpha)$

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Answer

To express the function in the required form, we start with the general cosine identity:

$R \text{cos}(x - \alpha) = R \text{cos} \alpha \text{cos} x + R \text{sin} \alpha \text{sin} x$ .

By comparing coefficients, we can set:

$R \text{cos} \alpha = 6$
$R \text{sin} \alpha = 2.5$

We can find $R$ using the Pythagorean identity:

$R = \sqrt{(6^2 + 2.5^2)} = \sqrt{36 + 6.25} = \sqrt{42.25} = 6.5$ .

Next, to find $\alpha$ , we use:

$\tan \alpha = \frac{\text{sin} \alpha}{\text{cos} \alpha} = \frac{2.5}{6}$ .

Calculating this gives:

$\alpha = \tan^{-1}\left(\frac{2.5}{6}\right) \approx 0.395.$

Thus, we express: $6 \text{cos} \, x + 2.5 \text{sin} \, x = 6.5 \text{cos}(x - 0.395)$ .

Step 2

Find the coordinates of the points on the graph where the curve $C$ crosses the coordinate axes

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Answer

To find where the curve crosses the axes, we set $y = 0$ for the x-axis intercepts:

$6 \text{cos} \, x + 2.5 \text{sin} \, x = 0.$

Using the cosine and sine relationship, we solve for $x$ . The values of $x$ when $y=0$ can be found at the points $x = 0.0, 1.97, 3.14, 5.10$ etc.

For the y-axis intercepts, we determine the value of $y$ when $x = 0$ :

$y = 6 \text{cos}(0) + 2.5 \text{sin}(0) = 6.$

Thus, the coordinates where the curve crosses are $(0, 6), (1.97, 0), (3.14, 0), (5.10, 0)$ .

Step 3

the maximum and minimum values of $H$ predicted by the model

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Answer

The function for $H$ is given by:

$H = 12 + 6 \text{cos} \left( \frac{2\pi t}{52} \right) + 2.5 \text{sin} \left( \frac{2\pi t}{52} \right).$

The maximum value of $H$ occurs when $\text{cos} \left( \frac{2\pi t}{52} \right) = 1$ and $\text{sin} \left( \frac{2\pi t}{52} \right) = 1$ :

$H_{max} = 12 + 6(1) + 2.5(1) = 18.5.$

The minimum value occurs when $\text{cos} \left( \frac{2\pi t}{52} \right) = -1$ and $\text{sin} \left( \frac{2\pi t}{52} \right) = -1$ :

$H_{min} = 12 + 6(-1) + 2.5(-1) = 5.5.$

Step 4

the values of $t$ when $H = 16$

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Answer

To find $t$ when $H = 16$ we substitute into the equation:

$16 = 12 + 6 \text{cos} \left( \frac{2\pi t}{52} \right) + 2.5 \text{sin} \left( \frac{2\pi t}{52} \right).$

Rearranging gives:

$4 = 6 \text{cos} \left( \frac{2\pi t}{52} \right) + 2.5 \text{sin} \left( \frac{2\pi t}{52} \right).$

Next, express this in terms of cosine and sine. This involves solving the trigonometric equation, leading to:

Use values for $t = 3$ or $t = 29$ when rounding to the nearest whole numbers.

Figure 1 shows the curve $C$, with equation $y = 6 \, ext{cos} \, x + 2.5 \, ext{sin} \, x$ for $0 \leq x \leq 2\pi$ - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 6

Worked Solution & Example Answer:Figure 1 shows the curve $C$, with equation $y = 6 \, ext{cos} \, x + 2.5 \, ext{sin} \, x$ for $0 \leq x \leq 2\pi$ - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 6

Express $6 \text{cos} \, x + 2.5 \text{sin} \, x$ in the form $R \text{cos} (x - \alpha)$

Find the coordinates of the points on the graph where the curve $C$ crosses the coordinate axes

the maximum and minimum values of $H$ predicted by the model

the values of $t$ when $H = 16$

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