12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sinx + 8 = 9 cos²x giving your answers to 2 decimal places - Edexcel - A-Level Maths Pure - Question 13 - 2017 - Paper 2

To solve the equation, start by using the identity for cosine: $ext{cos}^2 x = 1 - ext{sin}^2 x.$ Substituting this into the equation gives:

$3 ext{sin}^2 x + ext{sin} x + 8 = 9(1 - ext{sin}^2 x)$

which simplifies to:

$3 ext{sin}^2 x + ext{sin} x + 8 = 9 - 9 ext{sin}^2 x.$

Rearranging the terms, we combine like terms:

$12 ext{sin}^2 x + ext{sin} x - 1 = 0.$

This is a quadratic equation in terms of ( ext{sin} x). Using the quadratic formula:

$ext{sin} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where (a = 12), (b = 1), and (c = -1):

$ext{sin} x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 12 \cdot -1}}{2 \cdot 12}$

Calculating the discriminant:

$= \frac{-1 \pm \sqrt{1 + 48}}{24}= \frac{-1 \pm \sqrt{49}}{24}$

This simplifies to:

$= \frac{-1 \pm 7}{24},$

yielding two solutions:

1. (\text{sin} x = \frac{6}{24} = \frac{1}{4} \Rightarrow x = 14.48°)
2. (\text{sin} x = \frac{-8}{24} = -\frac{1}{3} \Rightarrow x = 165.52°, -19.47°, -160.53°.$$

Thus, the answers to two decimal places are:

1. (14.48°)
2. (165.52°)