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 Functions: 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 - 2009 - 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 - 2009 - 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 function in the required form, we first find the constants R and α.

Calculate R:

$R = \\sqrt{3^2 + 4^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5$
To find α, we use the tangent function:

$\tan(\alpha) = \frac{4}{3} \\Rightarrow \alpha = \tan^{-1}\left(\frac{4}{3}\right) \\approx 53^{\circ}$

Thus, we express the original equation as:

$3 \cos(\theta) + 4 \sin(\theta) = 5 \cos(\theta - 53^{\circ})$

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 $3 \cos(\theta) + 4 \sin(\theta)$ occurs when the cosine term is at its maximum, which is 1. Therefore, the maximum value is:

$5 \times 1 = 5$

To find the smallest positive value of θ, we set:

$\cos(\theta - 53^{\circ}) = 1$

This implies:

$\theta - 53^{\circ} = 0 \\Rightarrow \theta = 53^{\circ}$

Step 3

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

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Answer

Using the provided temperature model:

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

The minimum temperature occurs when the cosine term is at its minimum, which is -1:

$f(t) = 10 + 3(-1) + 4(0) = 10 - 3 = 7$

Thus, the minimum temperature of the warehouse is 7°C.

Step 4

Find the value of t when this minimum temperature occurs.

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Answer

We need to find when:

$\cos(15t) = -1$

This happens at:

t = 12 + 24k$$ For k = 0, this gives: $$t = 12$$ Thus, the value of t when the minimum temperature occurs is 12 hours after midday, which is 12:00 AM.

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 - 2009 - 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 - 2009 - 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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