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solve x in the triangle, round answer to the nearest tenth

To find the length of \( x \) in the right triangle provided in the image, we can use the sine function from trigonometry, which relates an angle of a right triangle to the ratio of the length of the opposite side and the hypotenuse.

Here, the given angle is \( 69^\circ \) and the side opposite this angle, which is also the hypotenuse in this context, is \( x \). The adjacent side to the angle is given as \( 5 \). We can use the tangent function since it involves the adjacent side and the opposite side (which we need to find).

The tangent of an angle \( \theta \) in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Here, the opposite side is \( x \) and the adjacent side is \( 5 \), and the angle \( \theta \) is \( 69^\circ \). So, we have:
\[
\tan(69^\circ) = \frac{x}{5}
\]

We solve for \( x \) by isolating it:
\[
x = 5 \times \tan(69^\circ)
\]

Now, let's compute the value of \( x \) using this equation.The length of \( x \) in the triangle is approximately \( 13.0 \) when rounded to the nearest tenth.

Would you like to explore this further or do you have any other questions?

Here are some related questions you might consider:

1. How would you calculate \( x \) if the given angle was \( 21^\circ \) instead of \( 69^\circ \)?
2. What if the length of the adjacent side was different, say \( 10 \) instead of \( 5 \)?
3. How would the calculation change if you were given the opposite side and needed to find the adjacent side?
4. What would happen to \( x \) if the angle increased to \( 80^\circ \)?
5. Can you solve for \( x \) using the sine function given a different hypotenuse length?
6. How do changes in the angle affect the length of \( x \)?
7. How would you determine the area of this triangle?
8. What is the relationship between the tangent and cotangent functions in this context?

**Tip:** When working with trigonometric functions, always ensure your calculator is set to the correct mode (degrees or radians) based on the units of the angle given.

Learn how to solve for the length of x in a right triangle with an angle of 69 degrees using the tangent function. This problem involves trigonometric concepts and is suitable for high school students.

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