what is the value of opposite side if tan theta is 30 degree and adjacant side is 12

To find the value of the opposite side, we can use the definition of the tangent function:

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

Given:
- \( \text{tan}(\theta) = 30^\circ \)
- \( \text{adjacent side} = 12 \)

We know that \( \text{tan}(30^\circ) = \frac{1}{\sqrt{3}} \).

Now, using the formula:

\[
\frac{1}{\sqrt{3}} = \frac{\text{opposite side}}{12}
\]

To solve for the opposite side:

\[
\text{opposite side} = 12 \times \frac{1}{\sqrt{3}} = \frac{12}{\sqrt{3}}
\]

We can rationalize the denominator:

\[
\text{opposite side} = \frac{12}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{12 \sqrt{3}}{3} = 4 \sqrt{3}
\]

Thus, the opposite side is \( 4\sqrt{3} \), which is approximately 6.93.

Would you like a detailed explanation or clarification on any part? 

Here are some related questions you might find helpful:
1. How do you calculate the hypotenuse using trigonometric ratios?
2. What is the exact value of \( \tan(30^\circ) \) and how is it derived?
3. Can you use sine or cosine to find the opposite side in this scenario?
4. How do you rationalize the denominator in trigonometric calculations?
5. How can the tangent function be used in real-world problems?

**Tip:** When dealing with square roots in fractions, always try to rationalize the denominator for cleaner answers.

Discover how to calculate the opposite side of a right triangle using tan 30° and an adjacent side length of 12. Learn the tangent function formula, rationalizing square roots, and apply it to find the opposite side as 4√3 (approximately 6.93). Suitable for students in Grades 9-11.

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