The problem consists of two parts: (a) Given $\cos x = 0.7431$ and $0^\circ < x < 90^\circ$, find the values of (i) $2\sin x$ and (ii) $\tan \frac{x}{2}$. We are instructed to use tables. Since we don't have tables available we will compute trigonometric values with calculator. (b) The interior angles of a pentagon are in the ratio $2:3:4:4:5$. Find the value of the largest angle.

1. Problem Description

The problem consists of two parts:

(a) Given $\cos x = 0.7431$ and $0^\circ < x < 90^\circ$, find the values of (i) $2\sin x$ and (ii) $\tan \frac{x}{2}$. We are instructed to use tables. Since we don't have tables available we will compute trigonometric values with calculator.

(b) The interior angles of a pentagon are in the ratio $2:3:4:4:5$. Find the value of the largest angle.

2. Solution Steps

(a)

Step 1: Find the value of $x$ using the given $\cos x = 0.7431$.

$x = \cos^{-1}(0.7431) \approx 42.0^\circ$

Step 2: Calculate $2\sin x$.

$2\sin x = 2\sin(42.0^\circ) \approx 2(0.6691) \approx 1.3382$

Step 3: Calculate $\tan \frac{x}{2}$.

$\tan \frac{x}{2} = \tan \frac{42.0^\circ}{2} = \tan(21.0^\circ) \approx 0.3839$

(b)

Step 1: The sum of the interior angles of a pentagon is given by the formula $(n-2) \times 180^\circ$, where $n$ is the number of sides. In this case, $n=5$.

Sum of interior angles $= (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$.

Step 2: Let the angles be $2k, 3k, 4k, 4k, 5k$ for some constant $k$. Then their sum is

$2k + 3k + 4k + 4k + 5k = 18k$.

Step 3: Since the sum of the angles is $540^\circ$, we have $18k = 540^\circ$. Therefore, $k = \frac{540^\circ}{18} = 30^\circ$.

Step 4: The largest angle is $5k = 5 \times 30^\circ = 150^\circ$.

3. Final Answer

(a) (i) $2\sin x \approx 1.3382$

(ii) $\tan \frac{x}{2} \approx 0.3839$

(b) The largest angle is $150^\circ$.