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In Figure 2 OAB is a sector of a circle, radius 5 m - Edexcel - A-Level Maths Pure - Question 6 - 2006 - Paper 2
Question 6
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... 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 6 - 2006 - 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:
[ c^2 = a^2 + b^2 - 2ab \cdot , cos(C) ]
Here, let:
- a = 5 m (radius OA)
- b = 5 m (radius OB)
- c = 6 m (chord AB)
Substituting the values, we get:
[ 6^2 = 5^2 + 5^2 - 2 \cdot 5 \cdot 5 \cdot cos(\angle AOB) ]
Simplifying this, [ 36 = 25 + 25 - 50 \cdot cos(\angle AOB) ]
[ 36 = 50 - 50 \cdot cos(\angle AOB) ]
[ 50 \cdot cos(\angle AOB) = 50 - 36 ]
[ cos(\angle AOB) = \frac{14}{50} = \frac{7}{25} ]
Step 2
Step 3
Calculate the area of the sector OAB.
Answer
The area of a sector can be calculated using the formula:
[ Area = \frac{1}{2} r^2 \theta ]
Where:
- r = radius = 5 m
- ( \theta = \angle AOB \approx 1.287 , ext{radians} )
Substituting the values:
[ Area = \frac{1}{2} \cdot 5^2 \cdot 1.287 \approx 16.087 , m^2 ]
Step 4
Hence calculate the shaded area.
Answer
To find the shaded area, we subtract the area of triangle OAB from the area of the sector OAB. The formula for the area of a triangle is:
[ Area_{triangle} = \frac{1}{2}ab \sin(C) ]
Using:
- a = 5 m
- b = 5 m
- C = ∠AOB ≈ 1.287 radians,
The area of triangle OAB is:
[ Area_{triangle} = \frac{1}{2} \cdot 5 \cdot 5 \cdot \sin(1.287) \approx 12 , m^2 ]
Hence, the shaded area is:
[ Shaded Area = Area_{sector} - Area_{triangle} \approx 16.087 - 12 \approx 4.087 , m^2 ]