9. (a) Express 2 sin θ - 4 cos θ in the form R sin(θ - α), where R and α are constants, R > 0 and 0 < α < π/2. Give the value of α to 3 decimal places. H(θ) = 4 + 5(2 sin 3θ - 4 cos 3θ)^2 Find (b) (i) the maximum value of H(θ), (ii) the smallest value of θ, for 0 ≤ θ < π, at which this maximum value occurs. (c) (i) the minimum value of H(θ), (ii) the largest value of θ, for 0 ≤ θ < π, at which this minimum value occurs.

A-Level Edexcel Maths Pure Rational Expressions: 9. (a) Express 2 sin θ

9. (a) Express 2 sin θ - 4 cos θ in the form R sin(θ - α), where R and α are constants, R > 0 and 0 < α < π/2 - Edexcel - A-Level Maths Pure - Question 1 - 2014 - Paper 5

Question 1

9. (a) Express 2 sin θ - 4 cos θ in the form R sin(θ - α), where R and α are constants, R > 0 and 0 < α < π/2. Give the value of α to 3 decimal places. H(θ) = 4 + ... show full transcript

Worked Solution & Example Answer:9. (a) Express 2 sin θ - 4 cos θ in the form R sin(θ - α), where R and α are constants, R > 0 and 0 < α < π/2 - Edexcel - A-Level Maths Pure - Question 1 - 2014 - Paper 5

Step 1

Express 2 sin θ - 4 cos θ in the form R sin(θ - α)

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Answer

To express the given expression in the form of R sin(θ - α), we begin with:

$R = ext{sqrt}(2^2 + (-4)^2) = ext{sqrt}(4 + 16) = ext{sqrt}(20) = 2 ext{sqrt}(5)$

The value of R is found to be approximately 4.472, which will be rounded off to 4.47.

Next, we calculate α using:

$an α = rac{-4}{2} = -2 \ ext{Thus, } α = an^{-1}(-2) \ ≈ 1.107 \ (63.43^ ext{o})$

Therefore, the values are:

R ≈ 4.472
α ≈ 1.107

Step 2

Find (b) (i) the maximum value of H(θ)

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Answer

To find the maximum value of H(θ), we first set the function which can be simplified as:

$H(θ) = 4 + 5(2 ext{sin}(3θ) - 4 ext{cos}(3θ))^2$

The term inside the square reaches its maximum when:

(equating to 6) \ (2 ext{sin}(3θ) = 6 + 4 ext{cos}(3θ)) $$ Thus the maximum value of H(θ) is 104 when simplifying using the critical points.

Step 3

Find (b) (ii) the smallest value of θ, for 0 ≤ θ < π, at which this maximum value occurs

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Answer

This occurs when:

$3θ ≈ 90^ ext{o} \ \therefore \ θ = rac{(90^ ext{o})}{3} = 30^ ext{o} \approx 0.524$

Step 4

Find (c) (i) the minimum value of H(θ)

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Answer

The minimum value of H(θ) occurs when:

The sum inside the squared term reaches its minimum, which yields:

$H_{min} = 4 + 5(0 - 4)^2 = 4 + 80 = 84$

Step 5

Find (c) (ii) the largest value of θ, for 0 ≤ θ < π, at which this minimum value occurs

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Answer

This occurs when:

$3θ = 360^ ext{o} \ \therefore \ θ = rac{360^ ext{o}}{3} = 120^ ext{o} \approx 2.094$

9. (a) Express 2 sin θ - 4 cos θ in the form R sin(θ - α), where R and α are constants, R > 0 and 0 < α < π/2 - Edexcel - A-Level Maths Pure - Question 1 - 2014 - Paper 5

Worked Solution & Example Answer:9. (a) Express 2 sin θ - 4 cos θ in the form R sin(θ - α), where R and α are constants, R > 0 and 0 < α < π/2 - Edexcel - A-Level Maths Pure - Question 1 - 2014 - Paper 5

Express 2 sin θ - 4 cos θ in the form R sin(θ - α)

Find (b) (i) the maximum value of H(θ)

Find (b) (ii) the smallest value of θ, for 0 ≤ θ < π, at which this maximum value occurs

Find (c) (i) the minimum value of H(θ)

Find (c) (ii) the largest value of θ, for 0 ≤ θ < π, at which this minimum value occurs

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