91Ó°ÊÓ

The angle sum property is a fundamental principle in geometry, especially relevant for triangles. This rule states that the sum of the internal angles in a triangle is always equal to \(180^\circ\) or \(\pi\) radians. This basic fact helps us determine an unknown angle when we already have two angles provided.
In the given triangle problem, we are given the measures of two angles: \(\angle B = \frac{\pi}{6}\) radians and \(\angle C = \frac{2\pi}{3}\) radians. To find the third angle \(\angle A\), we use the angle sum property. Simply subtract the sum of \(\angle B\) and \(\angle C\) from \(\pi\), the total degree measure of a triangle. Thus:

• \(\angle A = \pi - (\angle B + \angle C)\)
• \(\angle A = \pi - (\frac{\pi}{6} + \frac{2\pi}{3})\)
• Simplify the fractions to have a common denominator, making it \(\angle A = \frac{\pi}{6}\).

Knowing all the angles ensures that we can apply additional rules, like the law of sines, to solve for unknown sides.

Mathematics often uses radians as a way to measure angles, particularly in trigonometry and calculus. One important aspect of radian measure is its direct relationship to the unit circle, linking angles with trigonometric functions seamlessly.
A complete circle's circumference is \(2\pi\) times the radius, and this is equivalent to \(360^\circ\) degrees. Hence, \(\pi\) radians equal \(180^\circ\). This conversion is crucial for understanding and calculating angles in radians, simplifying many trigonometric equations.
For example:

• The given problem lists \(\angle C = \frac{2\pi}{3}\) radians, which corresponds to \(120^\circ\).
• Similarly, \(\angle B = \frac{\pi}{6}\) radians translates to \(30^\circ\) in degrees.

Using radian measure simplifies the use of trigonometric identities and calculus derivations, enhancing the fluidity of mathematical operations.

Rationalizing the denominator is a technique used to eliminate any irrational numbers, like square roots, from the denominator of a fraction. This process can make equations easier to handle and understand.
In our problem, after applying the law of sines, we determine that the side \(b\) is \(\frac{24}{\sqrt{3}}\). To rationalize the denominator:

• Multiply both the numerator and denominator by \(\sqrt{3}\).
• This results in \(\frac{24 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{24 \sqrt{3}}{3}\).
• Simplify the fraction to obtain \(8\sqrt{3}\).

This adjustment ensures the expression is in its simplest form, eliminating irrationality from the denominator, and often is preferred for final answers in mathematics.