Given that 5sinθ = 2cosθ, find the value of tanθ - Edexcel - A-Level Maths Pure - Question 7 - 2010 - Paper 3

Rearranging the equation gives:

$5 ext{sin}2x = 2 ext{cos}2x$

Divide by cos2x:

rac{5 ext{sin}2x}{ ext{cos}2x} = 2

This simplifies to:

$5 an 2x = 2$

Thus:

an 2x = rac{2}{5} = 0.4

To find the angles, we take:

$2x = an^{-1}(0.4)$

Calculating this gives:

$2x ext{ (using a calculator)} ext{ for the first quadrant: } 2x ext{ } ext{≈ } 21.8°$

Now, for the second quadrant, we can use:

$2x = 180° - 21.8° ext{ } ext{≈ } 158.2°$

Now we divide each angle by 2 to solve for x:

$x ext{ approximately equals: } 10.9°, 79.1°$

Next, we consider angles for 0 ≤ 2x < 360°:

$2x = 180° + 21.8° ext{ } ext{or } 2x = 360° - 21.8° ext{ } ext{giving us}$

$2x ≈ 201.8° ext{ and } 338.2°$

Thus:

$x ≈ 100.9°, 169.1°$

In conclusion, the solutions for x are:

• 10.9°
• 79.1°
• 100.9°
• 169.1°