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find the exact values of a and x with the final answer being a=40√3/3 and x=40/3

To solve for the exact values of \( a \) and \( x \) in the given right triangle, we can use trigonometric ratios based on the given angles of \( 30^\circ \) and \( 60^\circ \).

### Step 1: Understanding the Triangle
- The triangle is a right triangle.
- The angles are \( 30^\circ \), \( 60^\circ \), and \( 90^\circ \).
- The side opposite the \( 30^\circ \) angle is labeled \( x \) cm.
- The side opposite the \( 60^\circ \) angle is the given base \( 20 \) cm.
- The hypotenuse is labeled \( a \) cm.

### Step 2: Use the Trigonometric Ratios

#### To find \( x \):
Since \( x \) is opposite the \( 30^\circ \) angle and \( 20 \) cm is adjacent to the \( 30^\circ \) angle:
\[
\tan(30^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{20}
\]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[
\frac{1}{\sqrt{3}} = \frac{x}{20}
\]
\[
x = \frac{20}{\sqrt{3}} = \frac{20 \sqrt{3}}{3} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{20 \times \sqrt{3}}{3} \times \frac{1}{\sqrt{3}} = \frac{20}{3}
\]
Thus, \( x = \frac{20 \sqrt{3}}{3} \).

#### To find \( a \):
Since \( a \) is the hypotenuse:
\[
\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{20}{a}
\]
We know that \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), so:
\[
\frac{\sqrt{3}}{2} = \frac{20}{a}
\]
\[
a = \frac{20 \times 2}{\sqrt{3}} = \frac{40}{\sqrt{3}}
\]
Rationalizing the denominator:
\[
a = \frac{40 \times \sqrt{3}}{3}
\]

Thus, the exact values are:
\[
a = \frac{40\sqrt{3}}{3} \quad \text{and} \quad x = \frac{20}{\sqrt{3}} = \frac{20\sqrt{3}}{3}
\]

### Final Answer:
\[
a = \frac{40\sqrt{3}}{3}, \quad x = \frac{20\sqrt{3}}{3}
\]

Would you like any further details or have any questions?

### Relative Questions:
1. How does the sine and cosine of \(30^\circ\) and \(60^\circ\) relate to the side lengths in a 30-60-90 triangle?
2. What are the general properties of a 30-60-90 triangle?
3. How do you rationalize a denominator when dealing with square roots?
4. What would the side lengths be if the hypotenuse of the triangle was 50 cm instead of \(a\)?
5. How does the value of \( x \) change if the given adjacent side is increased?

### Tip:
Always remember that in a 30-60-90 triangle, the side opposite the \(30^\circ\) angle is always half the hypotenuse, and the side opposite the \(60^\circ\) angle is the hypotenuse times \(\sqrt{3}/2\).

Learn how to find the exact values of a and x in a 30-60-90 triangle using trigonometric ratios. Solve for a = 40√3/3 and x = 40/3 with a step-by-step explanation and trigonometric formulas. Suitable for high school students studying trigonometry.

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