A curve C has equation y = 3sin 2x + 4cos 2x, -π ≤ x ≤ π - Edexcel - A-Level Maths Pure - Question 8 - 2008 - Paper 6

To express y in the form Rsin(2x + α), we start with:

$y = 3sin(2x) + 4cos(2x)$

This can be rewritten using the sine addition formula. We find R as:

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

Next, we find α using:

$tan(α) = \frac{4}{3}$

Thus:

$α = arctan(\frac{4}{3}) ≈ 0.927$

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

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

This equation can be rearranged to:

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

Using the identity:

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

To solve for 2x:

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

This gives:

$x ≈ -1.02, 0.46, 1.11, 2.68$

Rounding to two decimal places:

$x ≈ -1.02, 0.46, 1.11, 2.68$