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solve for length x using trigonometric ratio

To solve for the length \( x \) in the right triangle using trigonometric ratios, we can use the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In this triangle:
- The angle given is \( 36^\circ \).
- The side opposite this angle is the side with length \( 18 \).
- The hypotenuse is the side we're trying to find, \( x \).

The sine function is given by:

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

Substituting the given values:

\[
\sin(36^\circ) = \frac{18}{x}
\]

Now, solve for \( x \):

\[
x = \frac{18}{\sin(36^\circ)}
\]

Let's calculate this value.

\[
\sin(36^\circ) \approx 0.5878
\]

\[
x \approx \frac{18}{0.5878} \approx 30.62
\]

Thus, the length \( x \) is approximately \( 30.62 \) units.

Would you like further details, or do you have any questions?

Here are some related questions you might consider:
1. How do you use the cosine function to find the adjacent side in this triangle?
2. What is the length of the adjacent side using the cosine function?
3. Can we verify the length \( x \) using the Pythagorean theorem?
4. How would you find the area of this right triangle?
5. What is the value of the other non-right angle in this triangle?
6. How does the value of \( \sin(36^\circ) \) change if the angle increases slightly?
7. What would be the length of \( x \) if the angle was \( 45^\circ \) instead of \( 36^\circ \)?
8. How do trigonometric ratios change when switching between different types of triangles?

**Tip:** Always ensure your calculator is in the correct mode (degrees or radians) when solving trigonometric problems to avoid calculation errors.

Learn how to solve for the length x in a right triangle using trigonometric ratios. This problem involves calculating the length of the hypotenuse using the sine function, given specific angle and opposite side lengths. Suitable for high school students.

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