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Figure 2 OAB is a sector of a circle, radius 5 m - Edexcel - A-Level Maths Pure - Question 8 - 2005 - Paper 2
Question 8
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, givi... show full transcript
Worked Solution & Example Answer: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 OAB:
Let (O) be the center of the circle, and (A) and (B) be points on the circumference. According to the cosine rule:
[ c^2 = a^2 + b^2 - 2ab \cos C ]
In triangle OAB:
- (c = AB = 6 m)
- (a = OB = 5 m)
- (b = OA = 5 m)
This gives us: [ 6^2 = 5^2 + 5^2 - 2 \cdot 5 \cdot 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} ]
Thus, we have shown that (\cos AOB = \frac{7}{25}.
Step 2
Step 3
Calculate the area of the sector OAB.
Answer
The area (A) of a sector can be calculated using the formula:
[ A = \frac{1}{2} r^2 \theta ]
where (r) is the radius and (\theta) is the angle in radians.
Here, (r = 5 m) and (\theta \approx 1.318 \text{ radians}), so:
[ A = \frac{1}{2} \cdot 5^2 \cdot 1.318 \approx 16.475 \text{ m}^2. ]
Step 4
Hence calculate the shaded area.
Answer
The shaded area can be calculated by subtracting the area of triangle OAB from the area of the sector OAB.
First, we calculate the area of triangle OAB using the formula:
[ \text{Area} = \frac{1}{2} ab \sin C ]
Here, (a = 5 m), (b = 5 m), and (C = AOB \approx 1.318), so:
[ \text{Area} = \frac{1}{2} \cdot 5 \cdot 5 \cdot \sin(1.318) \approx 12.831 \text{ m}^2. ]
Thus, the shaded area is: [ \text{Shaded Area} = \text{Area of Sector} - \text{Area of Triangle} \approx 16.475 - 12.831 \approx 3.644 \text{ m}^2. ]