Given that tan x = $\frac{2}{3}$, where 0o d" x d" 90o, Find the value of 2sinx.

1. Analyze the Question

We are given $\tan x = \frac{2}{3}$ in the first quadrant ($0^\circ \leq x \leq 90^\circ$). We need to find the value of $2\sin x$.

2. Explain the Concept

In a right-angled triangle, trigonometric ratios are defined by the mnemonic SOH CAH TOA:

• SOH: $\sin x = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• TOA: $\tan x = \frac{\text{Opposite}}{\text{Adjacent}}$

Since $\tan x = \frac{2}{3}$, we can treat the opposite side as $2$ and the adjacent side as $3$. We must first find the hypotenuse using Pythagoras' theorem:

3. Show the Solution

Step 1: Find the hypotenuse.

Step 2: Find $\sin x$. Using the definition $\sin x = \frac{\text{Opposite}}{\text{Hypotenuse}}$:

To rationalize the denominator:

Step 3: Calculate $2\sin x$.

4. Identify the Answer

Comparing our result to the given options, the correct value is $\frac{4\sqrt{13}}{13}$. This corresponds to option C.

5. Exam Tip

Whenever you are given one trigonometric ratio (like $\tan x$), always sketch a quick right-angled triangle. Label the sides based on the ratio, calculate the missing side using Pythagoras' theorem, and you can easily find any other ratio ($\sin x, \cos x, \sec x$, etc.) without getting confused!