Find the unknown side of the triangle.

Solution:

The image shows a right-angled triangle with one angle measuring 60 degrees. The side adjacent to the 60-degree angle and opposite the right angle has a length of 6. The unknown side, labeled 'x', is opposite the 60-degree angle.

We can use the trigonometric function tangent to find the unknown side:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
• In this triangle, \( \theta = 60^{\circ} \), the opposite side is \( x \), and the adjacent side is \( 6 \).
• So, \( \tan(60^{\circ}) = \frac{x}{6} \).
• We know that \( \tan(60^{\circ}) = \sqrt{3} \).
• Therefore, \( \sqrt{3} = \frac{x}{6} \).
• Multiplying both sides by 6, we get \( x = 6\sqrt{3} \).

Alternatively, we can use the sine function for the 60-degree angle and the hypotenuse, but we don't know the hypotenuse. We can find the hypotenuse using cosine:

• \( \cos(60^{\circ}) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \frac{1}{2} = \frac{6}{\text{hypotenuse}} \)
• Hypotenuse = \( 12 \)
• Now, using sine: \( \sin(60^{\circ}) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \frac{\sqrt{3}}{2} = \frac{x}{12} \)
• \( x = 12 \cdot \frac{\sqrt{3}}{2} = 6\sqrt{3} \)

The third angle in the triangle is \( 180^{\circ} - 90^{\circ} - 60^{\circ} = 30^{\circ} \). The side 'x' is opposite the 60-degree angle, and the side '6' is opposite the 30-degree angle.

• Using the ratio of sides in a 30-60-90 triangle: The side opposite the 60-degree angle is \( \sqrt{3} \) times the side opposite the 30-degree angle.
• So, \( x = 6 \cdot \sqrt{3} \).

Ответ: x = 6\(\sqrt{3}\).