Problem 6 - Algebra 2 and Trigonometry

Question

In \(\triangle A B C,\) if \(b=8, \mathrm{m} \angle A=\frac{\pi}{3},\) and \(\mathrm{m} \angle C=\frac{5 \pi}{12},\) find the exact value of \(a\) in simplest form.

Step-by-Step Solution

Answer

The exact value of \( a \) is \( 4\sqrt{6} \).

Step 1: Understanding the Law of Sines

The Law of Sines relates the ratio of each side of a triangle to the sine of its opposite angle. It is given by:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] In this problem, we need to find side \( a \) of \( \triangle ABC \). We will use the Law of Sines.

Step 2: Determining Angle B

First, calculate \( \angle B \) using the fact that the sum of angles in a triangle is \( \pi \) radians (180 degrees). The equation is:\[ \angle B = \pi - \angle A - \angle C = \pi - \frac{\pi}{3} - \frac{5\pi}{12} \] Find a common denominator and solve:

Step 3: Simplifying the Angle B Calculation

To solve: \( \pi = \frac{12\pi}{12}, \frac{\pi}{3} = \frac{4\pi}{12}, \frac{5\pi}{12} = \frac{5\pi}{12} \). Thus,\[ \angle B = \frac{12\pi}{12} - \frac{4\pi}{12} - \frac{5\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4} \] So \( \angle B = \frac{\pi}{4} \).

Step 4: Applying the Law of Sines

Now that we know all the angles, we can apply the Law of Sines:\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]Substitute \( A = \frac{\pi}{3} \), \( b = 8 \), and \( B = \frac{\pi}{4} \):

Step 5: Solving for Side a

Using \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), the equation becomes:\[ \frac{a}{\frac{\sqrt{3}}{2}} = \frac{8}{\frac{\sqrt{2}}{2}} \]Cross-multiply to solve for \( a \):

Step 6: Simplification

Cross-multiply to get:\[ a \cdot \frac{\sqrt{2}}{2} = 8 \cdot \frac{\sqrt{3}}{2} \]Multiply both sides by 2:\[ a \cdot \sqrt{2} = 8 \sqrt{3} \]Finally, divide both sides by \( \sqrt{2} \):

Step 7: Final Division and Result

The result is:\[ a = \frac{8 \sqrt{3}}{\sqrt{2}} \]Rationalize the denominator:\[ a = \frac{8 \sqrt{3}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{8 \sqrt{6}}{2} = 4 \sqrt{6} \]Thus, \( a = 4 \sqrt{6} \).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when working with triangles. These identities are mathematical equations that involve trigonometric functions such as sine, cosine, and tangent. In the problem involving triangle ABC, we utilize the Law of Sines, which is a trigonometric identity connecting the sides of a triangle with the sines of its angles. This identity is represented by the equation:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

This equation tells us that each side of the triangle divided by the sine of its opposite angle is equal. This property is very useful in solving for unknown sides or angles when some information is already known. For instance, if two angles and one side of a triangle are known, the Law of Sines helps us find the remaining unknowns. Remember, knowing the value of trigonometric functions for specific angles, like \( \sin \frac{\pi}{3} \) and \( \sin \frac{\pi}{4} \) is critical, as they help in solving these kinds of problems effectively. This is where trigonometric identities really shine.

Angle Sum in Triangle

A fundamental geometric rule about triangles is that the sum of the interior angles must always equal \( \pi \) radians, or 180 degrees. This is a key principle used in the solution to determine the unknown angle, \( \angle B \), in triangle ABC.

• The formula is: \( \angle A + \angle B + \angle C = \pi \)

In our exercise, angles \( A \) and \( C \) are given, and their measures are \( \frac{\pi}{3} \) and \( \frac{5\pi}{12} \) respectively. To find \( \angle B \), we subtract the sum of angles \( \angle A \) and \( \angle C \) from \( \pi \).

• Step-by-step, it's solved as: \( \angle B = \pi - \frac{\pi}{3} - \frac{5\pi}{12} \)

Finding a common denominator helps in performing the subtraction easily, giving us the final result: \( \angle B = \frac{\pi}{4} \). This calculation confirms that the internal angles correctly add up to \( \pi \), ensuring consistency and correctness in the work.

Simplifying Radicals

When dealing with mathematical expressions, simplifying radicals is an important skill. In this exercise, we encounter the challenge of simplifying the expression for side \( a \) after using the Law of Sines. Most times, such simplifications are necessary to reach a more understandable solution or to present the result in its simplest form.

• Initially, we find \( a = \frac{8\sqrt{3}}{\sqrt{2}} \).

This expression can become simpler through a process called rationalizing the denominator. To rationalize, we multiply both the numerator and the denominator by \( \sqrt{2} \) to eliminate the square root from the denominator.

• This transformation turns \( \frac{8\sqrt{3}}{\sqrt{2}} \) into \( \frac{8\sqrt{3}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{8\sqrt{6}}{2} \).
• The expression further simplifies to \( 4\sqrt{6} \) after dividing 8 by 2.

This simplification not only makes the numerical answer neater but also helps in understanding the problem thoroughly by breaking down complex parts into understandable pieces. Simplifying radicals is a fundamental mathematics skill, clear and systematic for tackling various math problems.