Solve, for $0 \leq x < 180^\circ$, $\cos(3x - 10^\circ) = -0.4$, giving your answers to 1 decimal place - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 6

Using the cosine function, we have:
$3x - 10^\circ = \alpha$
This gives us:
$3x - 10^\circ = 113.58^\circ.$

Rearranging the equation provides:
$3x = 113.58^\circ + 10^\circ = 123.58^\circ.$
Now divide by 3:
$x = \frac{123.58^\circ}{3} = 41.19^\circ \approx 41.2^\circ.$

Since cosine is negative in the second quadrant, we consider:
$3x - 10^\circ = 360^\circ - \alpha = 360^\circ - 113.58^\circ = 246.42^\circ.$

Following the same method, we have:
$3x = 246.42^\circ + 10^\circ = 256.42^\circ.$
Then, divide by 3 to find x:
$x = \frac{256.42^\circ}{3} = 85.47^\circ \approx 85.5^\circ.$

Next, we check for any further solutions using:
$3x - 10^\circ = 360^\circ + \alpha = 360^\circ + 113.58^\circ = 473.58^\circ.$
This leads to:
$3x = 473.58^\circ + 10^\circ = 483.58^\circ.$
Consequently:
$x = \frac{483.58^\circ}{3} = 161.19^\circ \approx 161.2^\circ.$

The final answers for x in the interval $0 \leq x < 180^\circ$ are:

• $x \approx 41.2^\circ$
• $x \approx 85.5^\circ$
• $x \approx 161.2^\circ$.