To find the hypotenuse, we'll use the definition of the sine function in a right triangle. The sine of an angle $\Theta$ is given by:

$\sin \Theta = \frac{\text{opposite}}{\text{hypotenuse}}$

Given:

• $\sin \Theta = \frac{1}{4}$
• Adjacent leg = 3 cm

Let's denote the hypotenuse as $h$ and the opposite leg as $o$.

Using the sine function:

$\frac{o}{h} = \frac{1}{4}$

This means:

$o = \frac{h}{4}$

Since we are not given the opposite leg, we can use the Pythagorean theorem, which states:

$h^2 = o^2 + \text{adjacent}^2$

Substitute $o = \frac{h}{4}$ and the adjacent leg:

$h^2 = \left(\frac{h}{4}\right)^2 + 3^2$

$h^2 = \frac{h^2}{16} + 9$

Now, solve for $h$:

$16h^2 = h^2 + 144$

$15h^2 = 144$

$h^2 = \frac{144}{15}$

$h^2 = 9.6$

$h = \sqrt{9.6}$

$h \approx 3.1 \, \text{cm}$

So, the hypotenuse is approximately $3.1$ cm.

Would you like more details or have any questions about this solution?

Here are 5 related questions:

1. How would you find the length of the opposite leg using the sine function?
2. What if the adjacent leg was 4 cm instead of 3 cm, how would the hypotenuse change?
3. How can the cosine function be used to find the hypotenuse?
4. Can you verify the result by recalculating using the cosine of the angle?
5. What is the exact length of the opposite leg using the value of the hypotenuse?

Tip: Always double-check your answers by verifying with the Pythagorean theorem when working with right triangles.