8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ − α), where R and α are constants, R > 0 and 0 < α < 90°. (b) Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs. The temperature, f(t), of a warehouse is modelled using the equation f(t) = 10 + 3 cos(15t°) + 4 sin(15t°), where t is the time in hours from midday and 0 ≤ t < 24. (c) Calculate the minimum temperature of the warehouse as given by this model. (d) Find the value of t when this minimum temperature occurs.

A-Level Edexcel Maths Pure Exponential & Logarithms: 8. (a) Express 3 cos θ + 4 sin θ

8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ − α), where R and α are constants, R > 0 and 0 < α < 90° - Edexcel - A-Level Maths Pure - Question 1 - 2008 - Paper 2

Question 1

8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ − α), where R and α are constants, R > 0 and 0 < α < 90°. (b) Hence find the maximum value of 3 cos θ + 4 sin θ... show full transcript

Worked Solution & Example Answer:8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ − α), where R and α are constants, R > 0 and 0 < α < 90° - Edexcel - A-Level Maths Pure - Question 1 - 2008 - Paper 2

Step 1

Express 3 cos θ + 4 sin θ in the form R cos(θ − α)

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Answer

To express the equation in the required form, we identify constants R and α. We start by calculating R:

$R = ext{sqrt}(3^2 + 4^2) = ext{sqrt}(9 + 16) = ext{sqrt}(25) = 5.$

Next, we find α using:

$an(α) = \frac{4}{3}$

From this, we calculate:

$α = an^{-1}\left(\frac{4}{3}\right) = 53°.$

Thus, we express the equation as:

$3 cos θ + 4 sin θ = 5 cos(θ − 53°).$

Step 2

Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs.

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Answer

The maximum value of the expression occurs when:

$ext{cos}(θ − α) = 1.$

This gives us:

ightarrow θ = α.$$ Thus, substituting α gives us: $$θ = 53°. $$ The maximum value is: $$ R = 5.$$

Step 3

Calculate the minimum temperature of the warehouse as given by this model.

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Answer

To calculate the minimum temperature, we first rewrite the function:

$f(t) = 10 + 3 cos(15t°) + 4 sin(15t°).$

The minimum temperature occurs when:

$f(t) = 10 + 5 cos(15t − α),$

where the minimum value of cos is -1. Thus:

$ext{Minimum temperature} = 10 + 5(-1) = 10 - 5 = 5°.$

Step 4

Find the value of t when this minimum temperature occurs.

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Answer

To find the specific time t for the minimum temperature, set:

$15t − α = 180°.$

Substituting for α gives:

$15t − 53° = 180°.$

Solving for t yields:

ightarrow t = \frac{233}{15} = 15.5333 ext{ hours}.$$ Converting this to minutes gives us: $$t = 15 + (0.5333 × 60) ext{ minutes} = 15 + 32 = 15:32.$$

8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ − α), where R and α are constants, R > 0 and 0 < α < 90° - Edexcel - A-Level Maths Pure - Question 1 - 2008 - Paper 2

Worked Solution & Example Answer:8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ − α), where R and α are constants, R > 0 and 0 < α < 90° - Edexcel - A-Level Maths Pure - Question 1 - 2008 - Paper 2

Express 3 cos θ + 4 sin θ in the form R cos(θ − α)

Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs.

Calculate the minimum temperature of the warehouse as given by this model.

Find the value of t when this minimum temperature occurs.

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