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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 tangent function. The tangent 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 side adjacent to the angle.

In this triangle:
- The angle given is \( 40^\circ \).
- The side opposite this angle is the side with length \( x \).
- The side adjacent to this angle is the side with length \( 7 \).

The tangent function is given by:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Substituting the given values:

\[
\tan(40^\circ) = \frac{x}{7}
\]

Now, solve for \( x \):

\[
x = 7 \times \tan(40^\circ)
\]

Let's calculate this value.

\[
\tan(40^\circ) \approx 0.8391
\]

\[
x \approx 7 \times 0.8391 \approx 5.8737
\]

Thus, the length \( x \) is approximately \( 5.87 \) units.

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

Here are some related questions you might consider:
1. How can you use the sine function to find the hypotenuse in this triangle?
2. What is the length of the hypotenuse using the sine function?
3. Can we use the cosine function to find the length \( x \)?
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. What would be the length of \( x \) if the angle was \( 50^\circ \) instead of \( 40^\circ \)?
7. How does the tangent function change as the angle increases?
8. How can we verify the length \( x \) using the Pythagorean theorem?

**Tip:** It's essential to double-check your trigonometric calculations, especially when working with angles, to ensure accuracy in your results.

Learn how to solve for the length x in a right triangle using trigonometric ratios, specifically the tangent function. This problem involves calculating the length of an unknown side given an angle and a known adjacent side. Suitable for high school students.

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