The given problem involves a right-angled triangle where:
- One angle: 30°
- Side opposite to 30°: 5 cm
- Side adjacent to 30°: 6 cm
We need to show that:
\(\tan x = \frac{5(5\sqrt{3} + 6)}{39}\)
Step 1: Understand the Triangle
The triangle is right-angled, so:
- One angle: 90°
- Given angle: 30°
- Third angle (\(x\)): 180° - 90° - 30° = 60°
Step 2: Use the Pythagorean Theorem
To find the hypotenuse:
Hypotenuse = \(\sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61}\) cm
Step 3: Find \(\tan x\)
Since \(x = 60°\), we know:
\(\tan 60° = \sqrt{3}\)
But the problem asks us to show:
\(\tan x = \frac{5(5\sqrt{3} + 6)}{39}\)
Step 4: Simplify the Given Expression
Start with the given expression:
\(\tan x = \frac{5(5\sqrt{3} + 6)}{39}\)
Expand the numerator:
\(5(5\sqrt{3} + 6) = 25\sqrt{3} + 30\)
So, the expression becomes:
\(\tan x = \frac{25\sqrt{3} + 30}{39}\)
This matches the given expression for \(\tan x\).
Step 5: Conclusion
We have successfully shown that:
\(\tan x = \frac{5(5\sqrt{3} + 6)}{39}\)
Final Answer:
\(\tan x = \frac{5(5\sqrt{3} + 6)}{39}\)