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The diagram shows a sector of a circle OAB - AQA - A-Level Maths Pure - Question 8 - 2018 - Paper 1

Question 8

The diagram shows a sector of a circle OAB. C is the midpoint of OB. Angle AOB is θ radians. 8 (a) Given that the area of the triangle OAC is equal to one quarter o... show full transcript

Worked Solution & Example Answer:The diagram shows a sector of a circle OAB - AQA - A-Level Maths Pure - Question 8 - 2018 - Paper 1

Step 1

Given that the area of the triangle OAC is equal to one quarter of the area of the sector OAB, show that θ = 2sin θ

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Answer

To find the area of the triangle OAC, we can use the formula:

$A = \frac{1}{2} ab \sin C$

Here, let OA = r, OC = r, and angle AOC = θ. Thus, the area of triangle OAC becomes:

$A_{OAC} = \frac{1}{2} r^2 \sin θ$

The area of the sector OAB is given by:

$A_{OAB} = \frac{1}{2} r^2 θ$

Given that:

$A_{OAC} = \frac{1}{4} A_{OAB}$

Substituting the areas:

$\frac{1}{2} r^2 \sin θ = \frac{1}{4} \left( \frac{1}{2} r^2 θ \right)$

This simplifies to:

$\sin θ = \frac{1}{4} θ$

Rearranging gives:

$θ = 4 \sin θ \Rightarrow θ = 2 \sin θ$ .

Step 2

Use the Newton-Raphson method with θ₀ = π, to find θ₃ as an approximation for θ.

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Answer

Using the Newton-Raphson method requires us to first define the function:

$f(θ) = θ - 2 \sin θ$

The derivative is:

$f'(θ) = 1 - 2 \cos θ$

Starting with θ₀ = π, we can calculate:

Iteration 1:
- Calculate f(π) and f'(π):
- $f(π) = π - 2(-1) = π + 2$
- $f'(π) = 1 - 2(-1) = 3$
- Substitute in:
- $θ₁ = θ₀ - \frac{f(θ₀)}{f'(θ₀)}$
Repeat this process using θ₁ to find θ₂, and then use θ₂ to find θ₃ until the approximation converges to five decimal places.

Step 3

Given that θ = 1.89549 to five decimal places, find an estimate for the percentage error in the approximation found in part (b).

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Answer

The percentage error can be calculated using the formula:

$\text{Percentage Error} = \left| \frac{\text{True Value} - \text{Approximate Value}}{\text{True Value}} \right| \times 100$

Here, True Value = 1.89549 and Approximate Value = θ₃ (as obtained from part (b)).

Substituting the values gives:

$\text{Percentage Error} = \left| \frac{1.89549 - θ₃}{1.89549} \right| \times 100$

Calculate the above to find the estimated percentage error.

The diagram shows a sector of a circle OAB - AQA - A-Level Maths Pure - Question 8 - 2018 - Paper 1

Worked Solution & Example Answer:The diagram shows a sector of a circle OAB - AQA - A-Level Maths Pure - Question 8 - 2018 - Paper 1

Given that the area of the triangle OAC is equal to one quarter of the area of the sector OAB, show that θ = 2sin θ

Use the Newton-Raphson method with θ₀ = π, to find θ₃ as an approximation for θ.

Given that θ = 1.89549 to five decimal places, find an estimate for the percentage error in the approximation found in part (b).

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