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Figure 3 shows a sketch of the curve with equation y = \frac{2 \sin 2x}{1 + \cos x} \quad 0 < x < \frac{\pi}{2} - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 8
Question 8
Figure 3 shows a sketch of the curve with equation y = \frac{2 \sin 2x}{1 + \cos x} \quad 0 < x < \frac{\pi}{2}. The finite region R, shown shaded in Figure 3, is ... show full transcript
Worked Solution & Example Answer:Figure 3 shows a sketch of the curve with equation y = \frac{2 \sin 2x}{1 + \cos x} \quad 0 < x < \frac{\pi}{2} - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 8
Step 1
Complete the table above giving the missing value of y to 5 decimal places.
Answer
To find the missing value of y when x = \frac{3\pi}{8}, we substitute this value into the equation:
Calculating the sine and cosine values, we find:
Thus, the completed table is:
| x | 0 | \frac{\pi}{8} | \frac{\pi}{4} | \frac{3\pi}{8} | \frac{\pi}{2} |
|---|---|---|---|---|---|
| y | 0 | 1.17157 | 1.02280 | 1.15073 |
Step 2
Use the trapezium rule with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 4 decimal places.
Answer
Applying the trapezium rule:
Area = \frac{h}{2} \left(y_0 + 2y_1 + 2y_2 + 2y_3 + y_4\right) ,
where h is the width of each interval.
Here, h = \frac{\pi/2 - 0}{4} = \frac{\pi}{8}.
Thus,
Area = \frac{\frac{\pi}{8}}{2} \left(0 + 2(1.17157) + 2(1.02280) + 2(1.15073) + 0\right) \approx 0.73508.
Final answer: Area \approx 0.7351.
Step 3
Using the substitution u = 1 + cos x; or otherwise, show that ∫ \frac{2 \sin 2x}{(1 + \cos x)} \, dx = 4 \ln(1 + \cos x) - 4 \cos x + k where k is a constant.
Answer
Using the substitution ( u = 1 + \cos x ) implies that ( du = -\sin x , dx ). After integration by parts and re-substituting, we arrive at:
Step 4
Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures.
Answer
To find the error, we calculate the area using the exact integration method and then find the difference with the trapezium rule estimate:
Error = Actual Area - Estimated Area.
Assuming the actual area calculated is approximately 1.150 to 2 decimal places, the error is:
Thus, the error to 2 significant figures is approximately 0.08.