6. (a) Show that the equation tan 2x = 5 sin 2x can be written in the form (1 - 5 cos 2x) sin 2x = 0 (b) Hence solve, for 0 ≤ x ≤ 180°, tan 2x = 5 sin 2x giving your answers to 1 decimal place where appropriate - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 3

Using the factored form from part (a):

$\sin 2x = 0 \quad \text{or} \quad 1 - 5 \cos 2x = 0$

For $\sin 2x = 0$:

This happens when:

$2x = n\pi$

For $0 ≤ x ≤ 180°$, the relevant solutions are:

$x = 0°, 90°$

Now for the second equation, setting $\cos 2x = \frac{1}{5}$ gives:

$2x = \cos^{-1}(\frac{1}{5})$

Calculating this:

$2x \approx 78.46° \implies x \approx 39.23°$

And also accounting for the second solution in the $0 ≤ 180°$ range:

$2x = 360° - \cos^{-1}(\frac{1}{5}) \approx 281.54° \implies x \approx 140.77°$

Hence, the final solutions, rounded to one decimal place, are:

• $x \approx 39.2°$
• $x \approx 140.8°$.