Determining the Characteristics of Triangle ABC with Given Angles and Sides

Given the triangle ABC with angle A 30 degrees, side b 8, and side a 6, we can determine the remaining angles and side. This process involves the application of trigonometric principles, primarily the Law of Sines. Let's explore this in detail.

Introduction to the Law of Sines

The Law of Sines, a fundamental principle in trigonometry, relates the lengths of the sides of a triangle to the sines of its angles. It is given by the equation:

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

Step 1: Applying the Law of Sines

We start by substituting the given values into the Law of Sines equation:

(frac{6}{sin30^circ} frac{8}{sin B})

Since (sin30^circ frac{1}{2}), we have:

(frac{6}{frac{1}{2}} frac{8}{sin B})

This simplifies to:

(12 frac{8}{sin B})

Step 2: Solving for (sin B)

Rearrange the equation to solve for (sin B):

(sin B frac{8}{12} frac{2}{3})

Step 3: Finding Angle B

To find angle B, we use the inverse sine function:

(B arcsinleft(frac{2}{3}right) approx 41.81^circ)

Step 4: Finding Angle C

Using the triangle angle sum property, where the sum of the angles in a triangle is 180 degrees:

(C 180^circ - A - B)

Substituting the known angles:

(C 180^circ - 30^circ - 41.81^circ approx 108.19^circ)

Step 5: Finding Side c

Using the Law of Sines again to find side c:

(frac{c}{sin C} frac{a}{sin A})

Substituting the known values:

(frac{c}{sin108.19^circ} frac{6}{sin30^circ})

This simplifies to:

(frac{c}{sin108.19^circ} 12)

Calculate (sin108.19^circ) which is approximately 0.9511:

(c approx 12 cdot 0.9511 approx 11.41)

Summary of Triangle ABC

The characteristics of triangle ABC are as follows:

Angle A 30 degrees Angle B (approx 41.81) degrees Angle C (approx 108.19) degrees Side a 6 Side b 8 Side c (approx 11.41)

Thus, triangle ABC is defined by the angles and sides calculated above.

Additional Calculations

Let's further elaborate on the additional calculations provided.

Angle A 41.81 degrees, Angle C 108.19 degrees, and side c 11.41

These values confirm our previous calculations.

Area of Triangle ABC Using Heron's Formula

To find the area of triangle ABC using Heron's formula, we first calculate the semi-perimeter:

(s frac{6 8 11.41}{2} 12.705)

Then, applying Heron's formula:

(text{Area} sqrt{s(s-a)(s-b)(s-c)} sqrt{12.705(12.705-6)(12.705-8)(12.705-11.41)} approx 22.8 text{ square units})

The sides of the triangle are:

Side a 6 units Side b 8 units Side c 11.4 units

The calculated area of 22.8 square units confirms the accuracy of our triangle characteristics.

Understanding the process of determining the characteristics of a triangle with given angles and sides is crucial for solving geometric problems and has practical applications in various fields, including architecture, engineering, and computer graphics.