A curve C has equation $y = 3 ext{sin}2x + 4 ext{cos}2x, - ext{π} < x < ext{π}$ - Edexcel - A-Level Maths Pure - Question 7 - 2008 - Paper 6

To express $y = 3\text{sin}(2x) + 4\text{cos}(2x)$ in the desired form, we need to find R and α:

We compute:

$R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Next, we find α:

$\tan(\alpha) = \frac{4}{3}$

This gives:

$\alpha = \arctan\left(\frac{4}{3}\right) \approx 0.927$ (to 3 significant figures).

The points of intersection with the x-axis occur where y = 0:

Setting the equation to zero gives:

$3\sin(2x) + 4\cos(2x) = 0$

This can be rearranged as:

$\sin(2x) = -\frac{4}{3}\cos(2x)$

Taking the sine of both sides yields:

$\tan(2x) = -\frac{4}{3}$

Finding values of 2x using:

$2x = \arctan\left(-\frac{4}{3}\right) + n\pi$

Calculating the values yields:

For 2x:

$2x \approx -2.03, 0.46, 3.11, 2.68$

Thus, the x-coordinates are:

$x \approx -1.015, 0.23, 1.555, 1.34$ (to 2 decimal places).