The image shows a right triangle with an angle of $27^\circ$, the hypotenuse of length 13, and the side opposite the angle $x$. To solve for $x$, we can use the sine function, since the sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse:

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

For this triangle:

• $\theta = 27^\circ$
• Opposite side = $x$
• Hypotenuse = 13

Thus, we set up the equation:

$\sin(27^\circ) = \frac{x}{13}$

Now, solve for $x$:

$x = 13 \times \sin(27^\circ)$

Using a calculator:

$x \approx 13 \times 0.4540 = 5.902$

So, $x \approx 5.9$.

Would you like more details on this or have any other questions?

Here are 5 related questions for further exploration:

1. How would you solve for the length of the adjacent side in this triangle?
2. What other trigonometric ratios could be used with this triangle?
3. How can you verify this solution using the Pythagorean theorem?
4. How does changing the angle affect the length of the opposite side?
5. What are the practical applications of using trigonometric ratios in real-life problems?

Tip: When working with trigonometry, it's essential to use your calculator in degree mode unless specified otherwise.