To solve for $x$ in the given right triangle, we can use trigonometric functions. Let's break this down:

Step 1: Identify the known values

• The hypotenuse ($c$) = $12 \, \text{cm}$
• The angle = $62^\circ$
• The side we need to find is $x$, which is the opposite side relative to the $62^\circ$ angle.

Step 2: Use the sine function

The sine function relates the opposite side and the hypotenuse: $\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}$

Substituting the values: $\sin(62^\circ) = \frac{x}{12}$

Step 3: Solve for $x$

$x = 12 \cdot \sin(62^\circ)$

Using a calculator: $\sin(62^\circ) \approx 0.8829$

So: $x = 12 \cdot 0.8829 = 10.5948$

Step 4: Round to the nearest hundredth

$x \approx 10.59 \, \text{cm}$

Final Answer:

$x = 10.59 \, \text{cm}$

Would you like me to explain this further or go into more detail?

Related Questions:

1. How do you use other trigonometric ratios (cosine, tangent) to solve for sides in a triangle?
2. How can you solve for the adjacent side using the same diagram?
3. What is the relationship between all three sides of a right triangle (Pythagorean theorem)?
4. How do you calculate angles in a triangle if all sides are known?
5. What are the differences between sine, cosine, and tangent in solving triangles?

Tip:

Always label the sides of a triangle (opposite, adjacent, hypotenuse) relative to the angle you’re working with. This helps you decide which trigonometric ratio to use!