To find the missing side $x$ in the given right triangle, where:

• The hypotenuse $c = 13$,
• The angle $\theta = 23^\circ$,
• $x$ is the side opposite to the $23^\circ$ angle.

We use the sine function, which relates the opposite side, hypotenuse, and the angle:

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

Substitute the known values:

$\sin(23^\circ) = \frac{x}{13}$

Now solve for $x$:

$x = 13 \cdot \sin(23^\circ)$

Step-by-Step Calculation:

1. Calculate $\sin(23^\circ)$: Using a calculator: $\sin(23^\circ) \approx 0.3907$

2. Multiply by the hypotenuse $13$: $x = 13 \cdot 0.3907 \approx 5.1$

Final Answer:

The missing side $x$ is approximately 5.1 units.

Let me know if you'd like a detailed breakdown of trigonometric functions or further clarification!

Related Questions:

1. How do you find the adjacent side using cosine in this triangle?
2. What is the tangent of $23^\circ$ in this triangle?
3. How do you solve for the angle if the opposite and adjacent sides are known?
4. Can you calculate the perimeter of the triangle?
5. How do you find the area of this triangle?

Tip: In right triangles, the sine function always relates the opposite side to the hypotenuse: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.