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12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sin x + 8 = 9 cos²x (b) Hence find the smallest positive solution of the equation 3sin(20° - 30°) + sin(20° - 30°) + 8 = 9 cos(20° - 30°) - Edexcel - A-Level Maths: Pure - Question 13 - 2017 - Paper 2
Question 13
12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sin x + 8 = 9 cos²x (b) Hence find the smallest positive solution of the equation 3sin(20° - 30°) + sin(... show full transcript
Worked Solution & Example Answer:12. (a) Solve, for -180° < x < 180°, the equation 3 sin²x + sin x + 8 = 9 cos²x (b) Hence find the smallest positive solution of the equation 3sin(20° - 30°) + sin(20° - 30°) + 8 = 9 cos(20° - 30°) - Edexcel - A-Level Maths: Pure - Question 13 - 2017 - Paper 2
Step 1
Solve, for -180° < x < 180°, the equation
Answer
To solve the equation, we can use the identity for cosine:
Substituting this into the equation gives:
Now we rearrange the equation:
Combining like terms results in:
We now have a quadratic in terms of , which we can solve using the quadratic formula:
ext{sin} x = rac{-b oxed{ ext{±}} ext{sqrt}(b^2 - 4ac)}{2a}
Where
- , , and .
Calculating the discriminant:
Substituting back, we find:
ext{sin} x = rac{-1 ext{±} 7}{24}
This gives us two values:
- ext{sin} x = rac{6}{24} = rac{1}{4}
- ext{sin} x = rac{-8}{24} = - rac{1}{3}
Using the inverse sine function to find the angles:
-
For ext{sin} x = rac{1}{4}:
-
For ext{sin} x = - rac{1}{3}:
Thus, the solutions are:
Step 2
Hence find the smallest positive solution of the equation
Answer
Starting from the previous calculation, we now need to find the smallest positive solution for the equation:
This simplifies to:
Substituting gives:
We then calculate:
Thus, the equation becomes:
Combining terms,
Rearranging gives:
To find the smallest positive solution, we calculate the angles. From the earlier analysis, we note:
ightarrow 2 ext{sin}(-10°) = - ext{sin}(19.47°) $$ Thus: $$ 2 ext{sin}(-10°) = -19.47° ightarrow heta = 5.26° $$