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The image shows a right triangle where:

- The hypotenuse is 10 cm.
- One leg (opposite to angle \( x \)) is 3 cm.
- The adjacent leg (denoted \( y \)) is not given directly.

We are asked to find the exact value of \( \tan(x) \).

### Solution:

The tangent of an angle \( \tan(x) \) in a right triangle is given by the ratio:

\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{y}
\]

To find \( y \), we can use the Pythagorean theorem since it's a right triangle:

\[
10^2 = y^2 + 3^2
\]

\[
100 = y^2 + 9
\]

\[
y^2 = 91
\]

\[
y = \sqrt{91}
\]

Now, we can calculate the exact value of \( \tan(x) \):

\[
\tan(x) = \frac{3}{\sqrt{91}}
\]

To rationalize the denominator:

\[
\tan(x) = \frac{3}{\sqrt{91}} \times \frac{\sqrt{91}}{\sqrt{91}} = \frac{3\sqrt{91}}{91}
\]

Thus, the exact value of \( \tan(x) \) is:

\[
\tan(x) = \frac{3\sqrt{91}}{91}
\]

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How do you find the sine and cosine of angle \( x \) in this triangle?
2. Can you find the area of this triangle using trigonometric ratios?
3. What are the possible applications of the tangent function in real-life problems?
4. How would you simplify the result if the opposite and adjacent legs had different values?
5. Can you solve for angle \( x \) in degrees or radians using this information?

**Tip**: When using trigonometric ratios, always check if you can simplify the expression by rationalizing denominators.

Write down the exact value of tan(x) for a right triangle with hypotenuse 10 cm, opposite side 3 cm, and unknown adjacent side y.

Discover how to calculate the exact value of tan(x) for a right triangle with a hypotenuse of 10 cm and an opposite side of 3 cm. The problem involves using trigonometric ratios and the Pythagorean theorem. Perfect for students in Grades 9-10.

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