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9. (a) Express sin θ − 2 cos θ in the form R sin(θ − α), where R > 0 and 0 < α < π/2 Give the exact value of R and the value of α, in radians, to 3 decimal places - Edexcel - A-Level Maths Pure - Question 2 - 2018 - Paper 5
Question 2
9. (a) Express sin θ − 2 cos θ in the form R sin(θ − α), where R > 0 and 0 < α < π/2 Give the exact value of R and the value of α, in radians, to 3 decimal places. ... show full transcript
Worked Solution & Example Answer:9. (a) Express sin θ − 2 cos θ in the form R sin(θ − α), where R > 0 and 0 < α < π/2 Give the exact value of R and the value of α, in radians, to 3 decimal places - Edexcel - A-Level Maths Pure - Question 2 - 2018 - Paper 5
Step 1
Express sin θ − 2 cos θ in the form R sin(θ − α)
Answer
To express ( sin θ − 2 cos θ ) in the form ( R sin(θ - α) ), we use the identity:
( R = \sqrt{A^2 + B^2} ) with ( A = 1 ) and ( B = -2 ).
Thus,
To determine ( α ), we use the relation:
( \tan(α) = \frac{B}{A} = \frac{-2}{1} = -2 )
This leads to:
( α = \tan^{-1}(-2) \approx 1.107 \text{ radians} (3 ext{ decimal places})$$
Step 2
Find (i) the maximum value of M(θ)
Answer
To find the maximum value of ( M(θ) = 40 + (3(sin θ - 6 cos^2 θ)^2) ), we start by rewriting:
( M(θ) = 40 + (3(sin θ - 6 \cdot \frac{1 - cos(2θ)}{2}))^2 ).
We evaluate the critical points where ( M(θ) ) reaches its maximum using calculus, yielding the maximum value as ( 85 ).
Step 3
Step 4
Step 5
Find (ii) the largest value of θ in the range 0 < θ < 2π at which the maximum value of N(θ) occurs
Answer
To identify the largest corresponding angle, we set:
( 2θ - 1.107 = nπ )
yielding ( θ = \frac{1.107 + nπ}{2} ). The valid integer values for n will yield acceptable values of θ, giving us approximately 5.27 as the solution.