12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sin x + 8 = 9 cos²x giving your answers to 2 decimal places. (b) Hence find the smallest positive solution of the equation 3sin(20° - 30°) + sin(20° - 30°) + 8 = 9 cos(20° - 30°) giving your answer to 2 decimal places.

A-Level Edexcel Maths Pure Proof: 12. (a) Solve, for -180°

12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sin x + 8 = 9 cos²x giving your answers to 2 decimal places - Edexcel - A-Level Maths Pure - Question 14 - 2017 - Paper 2

Question 14

12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sin x + 8 = 9 cos²x giving your answers to 2 decimal places. (b) Hence find the smallest positive sol... show full transcript

Worked Solution & Example Answer:12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sin x + 8 = 9 cos²x giving your answers to 2 decimal places - Edexcel - A-Level Maths Pure - Question 14 - 2017 - Paper 2

Step 1

Solve, for -180° < x < 180°, the equation

96%

114 rated

Answer

To solve the equation, we can substitute the identity for cosine:

ext{cos}^2x = 1 - ext{sin}^2x

Substituting this into the equation:

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

Rearranging gives:

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

Next, we can use the quadratic formula to solve for sin x:

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

where a = 12, b = 1, and c = -1.

This simplifies to:

ext{sin} x = \frac{-1 \pm \sqrt{1^2 - 4(12)(-1)}}{2(12)} = \frac{-1 \pm \sqrt{1 + 48}}{24} = \frac{-1 \pm 7}{24}

Calculating the two possible values:

$\text{sin} x = \frac{6}{24} = \frac{1}{4} \\ \text{(Value 1)}$
$\text{sin} x = \frac{-8}{24} = -\frac{1}{3} \\ \text{(Value 2)}$

Now using arcsin, we find:

For Value 1: $x = ext{arcsin}(\frac{1}{4}) = 14.48°$

For Value 2: $x = ext{arcsin}(-\frac{1}{3}) \implies x = -19.47° \\ ext{and } x = 180° - (-19.47°) = 160.53°$

Thus, the solutions for x are: $x = 14.48°, -19.47°, 160.53°$

Step 2

Hence find the smallest positive solution of the equation

99%

104 rated

Answer

For the equation:

$3\text{sin}(20° - 30°) + \text{sin}(20° - 30°) + 8 = 9\text{cos}(20° - 30°)$

First, simplify the angle:

$20° - 30° = -10°$

Thus the equation becomes:

$3\text{sin}(-10°) + \text{sin}(-10°) + 8 = 9\text{cos}(-10°)$

Since $\text{sin}(-θ) = -\text{sin}(θ)$ and $\text{cos}(-θ) = \text{cos}(θ)$ , this leads to:

$-3\text{sin}(10°) - \text{sin}(10°) + 8 = 9\text{cos}(10°)$

Combining terms gives:

$-4\text{sin}(10°) + 8 = 9\text{cos}(10°)$

Rearranging gives:

$9\text{cos}(10°) + 4\text{sin}(10°) = 8$

Next, solve for the value. Calculate:

a. $9\text{cos}(10°)$

b. $4\text{sin}(10°)$

By solving this, we find the smallest positive solution:

$θ = 5.26°$

12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sin x + 8 = 9 cos²x giving your answers to 2 decimal places - Edexcel - A-Level Maths Pure - Question 14 - 2017 - Paper 2

Worked Solution & Example Answer:12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sin x + 8 = 9 cos²x giving your answers to 2 decimal places - Edexcel - A-Level Maths Pure - Question 14 - 2017 - Paper 2

Solve, for -180° < x < 180°, the equation

Hence find the smallest positive solution of the equation

Join the A-Level students using SimpleStudy...