In the diagram, OMN is a sector of a circle with angle θ at the centre. OM = ON = 6 cm and MN = 8 cm. Find, correct to the nearest whole number, the value of θ. Calculate, correct to two significant figures, the area of the shaded region.

Here's the solution for question 9.

a) Step 1: Apply the Cosine Rule to find $\theta$.
Given $OM = 6$ cm, $ON = 6$ cm, and $MN = 8$ cm.
The cosine rule states $c^2 = a^2 + b^2 - 2ab \cos C$. In $\triangle OMN$, we have:

$$MN^2 = OM^2 + ON^2 - 2(OM)(ON) \cos \theta$$

Substitute the given values:

$$8^2 = 6^2 + 6^2 - 2(6)(6) \cos \theta$$ $$64 = 36 + 36 - 72 \cos \theta$$ $$64 = 72 - 72 \cos \theta$$

Rearrange to solve for $\cos \theta$:

$$72 \cos \theta = 72 - 64$$ $$72 \cos \theta = 8$$ $$\cos \theta = \frac{8}{72} = \frac{1}{9}$$

Calculate $\theta$:

$$\theta = \arccos\left(\frac{1}{9}\right)$$ $$\theta \approx 83.6206^\circ$$

Correct to the nearest whole number:

$$\theta = \boxed{84^\circ}$$

b) Step 2: Calculate the area of the shaded region.
The shaded region is the area of the sector $OMN$ minus the area of the triangle $OMN$.
Radius of the sector $r = 6$ cm.
Angle $\theta \approx 83.6206^\circ$.
Use $\pi = \frac{22}{7}$.

First, calculate the area of the sector $OMN$:

$$\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2$$ $$\text{Area of sector} = \frac{83.6206}{360} \times \frac{22}{7} \times (6)^2$$ $$\text{Area of sector} = \frac{83.6206}{360} \times \frac{22}{7} \times 36$$ $$\text{Area of sector} \approx 26.279 \text{ cm}^2$$

Next, calculate the area of the triangle $OMN$:

$$\text{Area of triangle} = \frac{1}{2} OM \cdot ON \sin \theta$$ $$\text{Area of triangle} = \frac{1}{2} (6)(6) \sin(83.6206^\circ)$$ $$\text{Area of triangle} = 18 \sin(83.6206^\circ)$$ $$\text{Area of triangle} \approx 18 \times 0.9937$$ $$\text{Area of triangle} \approx 17.8866 \text{ cm}^2$$

Finally, calculate the area of the shaded region:

$$\text{Area of shaded region} = \text{Area of sector} - \text{Area of triangle}$$ $$\text{Area of shaded region} = 26.279 - 17.8866$$ $$\text{Area of shaded region} = 8.3924 \text{ cm}^2$$

Correct to two significant figures:

$$\text{Area of shaded region} = \boxed{8.4 \text{ cm}^2}$$