In Figure 2 OAB is a sector of a circle, radius 5 m - Edexcel - A-Level Maths Pure - Question 7 - 2006 - Paper 2

Using the relationship between cosine and angle:

$AOB = \cos^{-1}(\frac{7}{25}).$

Calculating this gives:

$AOB \approx 1.287 \text{ radians}$

Thus the angle AOB in radians is approximately 1.287.

The area of a sector is given by the formula:

$\text{Area} = \frac{1}{2} \cdot r^2 \cdot \theta$

where ( r = 5 ) m and ( \theta \approx 1.287 \text{ radians} ):

$\text{Area} = \frac{1}{2} \cdot 5^2 \cdot 1.287 \approx 16.087 \text{ m}^2.$

To find the shaded area, we need to subtract the area of triangle OAB from the area of sector OAB. The area of triangle OAB can be calculated using:

$\text{Area of triangle} = \frac{1}{2} \cdot a \cdot b \cdot \sin \theta$

Substituting in:

• a = 5 m
• b = 5 m
• ( \theta \approx 1.287 \text{ radians} )

$\text{Area of triangle} = \frac{1}{2} \cdot 5 \cdot 5 \cdot \sin(1.287) \approx 12.$

Thus, the shaded area can be calculated as:

$\text{Shaded Area} = \text{Area of sector} - \text{Area of triangle} = 16.087 - 12 \approx 4.087 \text{ m}^2.$