6. (a) Express sin x + 2 cos x in the form R sin(x + a) where R and a are constants, R > 0 and 0 < a < π/2 Give the exact value of R and give the value of a in radians to 3 decimal places. (3) The temperature, θ°C, inside a room on a given day is modelled by the equation θ = 5 + sin(π/12 i - 3) + 2 cos(π/12 - 3) 0 ≤ i < 24 Using the equation of the model and your answer to part (a), (b) deduce the maximum temperature of the room during this day, (1) (c) find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.

A-Level Edexcel Maths Pure Graphs of Functions: 6. (a) Express sin x + 2 cos x i

6. (a) Express sin x + 2 cos x in the form R sin(x + a) where R and a are constants, R > 0 and 0 < a < π/2 Give the exact value of R and give the value of a in radians to 3 decimal places - Edexcel - A-Level Maths Pure - Question 7 - 2020 - Paper 1

Question 7

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6. (a) Express sin x + 2 cos x in the form R sin(x + a) where R and a are constants, R > 0 and 0 < a < π/2 Give the exact value of R and give the value of a in radia... show full transcript

Worked Solution & Example Answer:6. (a) Express sin x + 2 cos x in the form R sin(x + a) where R and a are constants, R > 0 and 0 < a < π/2 Give the exact value of R and give the value of a in radians to 3 decimal places - Edexcel - A-Level Maths Pure - Question 7 - 2020 - Paper 1

Step 1

Express sin x + 2 cos x in the form R sin(x + a)

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Answer

To express the equation in the form of R sin(x + a), we can start by finding the values of R and a:

First, calculate R: $R = \sqrt{1^2 + 2^2} = \sqrt{5}$
Next, find the angle a using the tangent function: $\tan a = \frac{2}{1} = 2 \Rightarrow a = \tan^{-1}(2) \approx 1.107 \text{ radians}$

Thus, the expression can be rewritten as: $R \sin(x + a) = \sqrt{5} \sin(x + 1.107)$

Step 2

Deduce the maximum temperature of the room during this day

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Answer

Using the expression for temperature: $θ = 5 + \sin(\frac{\pi}{12} i - 3) + 2 \cos(\frac{\pi}{12} - 3)$

To find the maximum temperature, we need to determine the maximum value of the sine and cosine functions. The maximum value of $\sin(x)$ is 1 and for $\cos(x)$ is also 1. Therefore:

Maximum contribution from sine: 1
Maximum contribution from cosine: 2

Thus, the maximum temperature can be calculated as: $θ_{max} = 5 + 1 + 2 = 8°C$

Step 3

Find the time of day when the maximum temperature occurs

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Answer

To find the time, we identify when the sine and cosine reach their maximum values:

For maximum $\sin(\frac{\pi}{12} i - 3)$ , set it to equal 1: $\frac{\pi}{12} i - 3 = \frac{\pi}{2} + 2k\pi$ Simplifying, we find: $\frac{\pi}{12} i = 3 + \frac{\pi}{2} + 2k\pi$ Solving for i gives multiple angles, but the first relevant is when $i = 13.2$ .
So the time is approximately: $i = 13.2 \Rightarrow 13:12 ext{ or } 1:12 PM$

Thus, the maximum temperature occurs at approximately 1:14 PM.

6. (a) Express sin x + 2 cos x in the form R sin(x + a) where R and a are constants, R > 0 and 0 < a < π/2 Give the exact value of R and give the value of a in radians to 3 decimal places - Edexcel - A-Level Maths Pure - Question 7 - 2020 - Paper 1

Worked Solution & Example Answer:6. (a) Express sin x + 2 cos x in the form R sin(x + a) where R and a are constants, R > 0 and 0 < a < π/2 Give the exact value of R and give the value of a in radians to 3 decimal places - Edexcel - A-Level Maths Pure - Question 7 - 2020 - Paper 1

Express sin x + 2 cos x in the form R sin(x + a)

Deduce the maximum temperature of the room during this day

Find the time of day when the maximum temperature occurs

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