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In Figure 2 OAB is a sector of a circle, radius 5 m - Edexcel - A-Level Maths Pure - Question 8 - 2005 - Paper 2
Question 8
In Figure 2 OAB is a sector of a circle, radius 5 m. The chord AB is 6 m long. (a) Show that cos AOB = \(\frac{7}{25}\) (b) Hence find the angle AOB in radians, gi... show full transcript
Worked Solution & Example Answer:In Figure 2 OAB is a sector of a circle, radius 5 m - Edexcel - A-Level Maths Pure - Question 8 - 2005 - Paper 2
Step 1
Show that cos AOB = \(\frac{7}{25}\)
Answer
To find (\cos AOB), we can use the cosine rule in triangle AOB:
Given:
- OA = OB = 5 m (radii of the circle)
- AB = 6 m
Applying the cosine rule:
[ c^2 = a^2 + b^2 - 2ab \cos(C) ]
- Here, (c = AB = 6), (a = OA = 5), and (b = OB = 5).
Substituting the values: [ 6^2 = 5^2 + 5^2 - 2(5)(5)\cos(AOB) ] [ 36 = 25 + 25 - 50\cos(AOB) ] [ 36 = 50 - 50\cos(AOB) ] [ 50\cos(AOB) = 50 - 36 ] [ 50\cos(AOB) = 14 ] [ \cos(AOB) = \frac{14}{50} = \frac{7}{25} ]
Step 2
Hence find the angle AOB in radians, giving your answer to 3 decimal places.
Answer
Given that (\cos AOB = \frac{7}{25}), we can find the angle AOB using the inverse cosine function:
[ AOB = \cos^{-1}\left(\frac{7}{25}\right) ]
Using a calculator, we find:
[ AOB \approx 1.151 \text{ radians (to 3 decimal places)} ]
Step 3
Calculate the area of the sector OAB.
Answer
The area of a sector can be calculated using the formula:
[ \text{Area} = \frac{1}{2}r^2\theta ]
where (r = 5) m and (\theta = 1.151) radians. Thus:
[ \text{Area} = \frac{1}{2}(5)^2(1.151) = \frac{1}{2} \times 25 \times 1.151 = 14.388 \text{ m}^2 ]
Step 4
Hence calculate the shaded area.
Answer
To find the shaded area, we need to subtract the area of triangle AOB from the area of the sector OAB.
The area of triangle AOB can be calculated by:
[ \text{Area of } \triangle AOB = \frac{1}{2} \times a \times b \times \sin(C) ]
where (a = 5), (b = 5), and (C = AOB \approx 1.151):
[ \text{Area} = \frac{1}{2} \times 5 \times 5 \times \sin(1.151) \approx \frac{25}{2} \times 0.887 = 11.088 \text{ m}^2 ]
Thus, the shaded area is:
[ \text{Shaded Area} = \text{Area of Sector OAB} - \text{Area of } \triangle AOB ] [ \text{Shaded Area} = 14.388 - 11.088 = 3.300 \text{ m}^2 ]