Solve the following equations, correct to one decimal place: (ii) 5 theta − 1 = 0, where 0 theta 90^

Here are the solutions to the problems:

c. Solve the following equations, correct to one decimal place: (ii) $5 \cos \theta - 1 = 0$, where $0 \le \theta \le 90^\circ$

Step 1: Isolate $\cos \theta$.

$$5 \cos \theta - 1 = 0$$ $$5 \cos \theta = 1$$ $$\cos \theta = \frac{1}{5}$$

Step 2: Find the value of $\theta$.

$$\theta = \cos^{-1}\left(\frac{1}{5}\right)$$ $$\theta \approx 78.463^\circ$$

Step 3: Round to one decimal place.

$$\theta \approx 78.5^\circ$$

The value $78.5^\circ$ is within the range $0^\circ \le \theta \le 90^\circ$.

The final answer is $\boxed{\text{78.5}^\circ}$.

11a. Calculate, correct to 3 s.f. the area of the minor segment cut-off by XY.

Step 1: Identify the given values.
Radius $r = 5$ cm.
Angle subtended at the centre $\theta = 96^\circ$.
$\pi = \frac{22}{7}$.

Step 2: Calculate the area of the sector OXY.
The formula for the area of a sector is $\frac{\theta}{360^\circ} \pi r^2$.

$$\text{Area of sector} = \frac{96}{360} \times \frac{22}{7} \times (5)^2$$ $$\text{Area of sector} = \frac{96}{360} \times \frac{22}{7} \times 25$$ $$\text{Area of sector} = \frac{440}{21} \approx 20.95238 \text{ cm}^2$$

Step 3: Calculate the area of the triangle OXY.
The formula for the area of a triangle with two sides and the included angle is $\frac{1}{2} r^2 \sin \theta$.

$$\text{Area of triangle} = \frac{1}{2} \times (5)^2 \times \sin(96^\circ)$$ $$\text{Area of triangle} = \frac{1}{2} \times 25 \times \sin(96^\circ)$$ $$\text{Area of triangle} \approx 12.5 \times 0.994521 \approx 12.43151 \text{ cm}^2$$

Step 4: Calculate the area of the minor segment.
Area of minor segment = Area of sector - Area of triangle.

$$\text{Area of segment} \approx 20.95238 - 12.43151$$ $$\text{Area of segment} \approx 8.52087 \text{ cm}^2$$

Step 5: Round the answer to 3 significant figures.

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