Solve, for -180° ≤ θ ≤ 180°, the equation 5 sin 2θ = 9 tan θ giving your answers, where necessary, to one decimal place - Edexcel - A-Level Maths Pure - Question 7 - 2019 - Paper 2

To find the smallest positive solution:

1. Rewrite the equation:

   • We again start with:

   $5 \sin(2x - 50°) = 9 \tan(x - 25°)$

   Using the double angle formula for sine:

   $5 \cdot 2 \sin(x - 25°) \cos(x - 25°) = 9 \frac{\sin(x - 25°)}{\cos(x - 25°)}$

2. Simplifying the relation:

   • This gives us a similar rearrangement:

   $10 \sin(x - 25°) \cos(x - 25°) = 9 \sin(x - 25°)$

   • Which leads to:

   $\sin(x - 25°)(10 \cos(x - 25°) - 9) = 0$

3. Finding solutions for each component:

   • For \sin(x - 25°) = 0, we have:
     • (x - 25° = n \cdot 180°,\text{ for } n \in \mathbb{Z} \Rightarrow x = 25° + n \cdot 180°.
   • For \10 \cos(x - 25°) - 9 = 0:
     • This implies \cos(x - 25°) = \frac{9}{10},\text{ yielding further solutions. }

4. Smallest positive solution:

   • The solution from \sin(x - 25°) = 0 gives us:
     • When n = 0, (x = 25°), and for higher n, we find larger values.
   • From \cos(x - 25°):
     • The first positive instance will also yield a solution that can be calculated and found.

Thus, the smallest positive solution is:

• x = 25°.