Problem 12 - Algebra 2 and Trigonometry

Question

In \(\triangle A B C, \mathrm{m} \angle A=102, \mathrm{m} \angle B=34,\) and \(a=25.8 .\) Find \(c\) to the nearest tenth.

Step-by-Step Solution

Answer

The length of \(c\) is 18.3.

Step 1: Understand the Angle Sum Property

In a triangle, the sum of all the interior angles is always 180 degrees. We have two of the angles given, \(\angle A = 102\) and \(\angle B = 34\). We need to find \(\angle C\) using the formula: \(\angle C = 180 - \angle A - \angle B\).

Step 2: Calculate \(\angle C\)

Substitute the known values into the angle sum formula: \(\angle C = 180 - 102 - 34\), which simplifies to \(\angle C = 44\) degrees.

Step 3: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A} = \frac{c}{\sin C}\). We already know \(a = 25.8\), \(\angle A = 102\) and \(\angle C = 44\). Our goal is to find \(c\).

Step 4: Setup the Equation Using the Law of Sines

Using the Law of Sines formula, substitute the known values: \(\frac{25.8}{\sin 102} = \frac{c}{\sin 44}\).

Step 5: Solve for \(c\)

First, calculate \(\sin 102\) and \(\sin 44\) using a calculator. Then substitute those values back into the equation: \(\frac{25.8}{0.9781} = \frac{c}{0.6946}\). Cross-multiply to get \(c = \frac{25.8 \times 0.6946}{0.9781}\).

Step 6: Calculate and Round \(c\)

Perform the calculation: \(c = \frac{25.8 \times 0.6946}{0.9781} \approx 18.3\). Round \(c\) to the nearest tenth to get \(c = 18.3\).

Key Concepts

Angle Sum Property

Every triangle has three corners or angles. When you add these three angles together, they always equal 180 degrees. This is known as the angle sum property of triangles. It's a crucial rule that helps us solve many problems involving triangles. For instance, if we know two angles of a triangle, we can easily find the third angle by subtracting the sum of the known angles from 180.

In our problem, we have been given two angles of triangle ABC: \( \angle A = 102 \) degrees and \( \angle B = 34 \) degrees. To find \( \angle C \), we use the angle sum property:

• Calculate the sum \( \angle A + \angle B = 102 + 34 = 136 \) degrees.
• Subtract this sum from 180: \( 180 - 136 = 44 \) degrees.

So, \( \angle C = 44 \) degrees. Understanding this property allows us to fully solve the triangle and find missing angles anytime the other angles are known.

Interior Angles of a Triangle

The angles inside a triangle are called interior angles. These angles help define the shape and type of the triangle. For example, if all three angles in a triangle are less than 90 degrees, it is an acute triangle. If one of the angles is exactly 90 degrees, it is a right triangle. If one of the angles is more than 90 degrees, it is an obtuse triangle.

In our problem, we have triangle ABC. Initially, we are given two angles, \( \angle A \) and \( \angle B \). Once we use the angle sum property, we find \( \angle C = 44 \) degrees. Since all angles in triangle ABC are less than 90 degrees, it is an acute triangle.

Understanding interior angles helps in many geometric and trigonometric calculations. It gives insight into the properties of the triangle and its classification. This basic knowledge plays a key role when we later apply laws, such as the Law of Sines or Cosines, to find missing elements in the triangle.

Solving Triangles

Solving a triangle means finding the missing parts of the triangle, usually the lengths of sides or the measures of angles. In trigonometry, various laws or rules are used to solve triangles. One such rule is the Law of Sines, which relates the sides of a triangle to its angles.

In our exercise, after finding \( \angle C \), we use the Law of Sines to find the missing side \( c \). The Law of Sines states:

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

For triangle ABC, with \( a = 25.8 \), \( \angle A = 102 \) degrees, and \( \angle C = 44 \) degrees:

• We set up our equation: \( \frac{25.8}{\sin 102} = \frac{c}{\sin 44} \)
• Using a calculator, find \( \sin 102 \approx 0.9781 \) and \( \sin 44 \approx 0.6946 \).
• Substitute these into the equation and solve for \( c \): \( c = \frac{25.8 \times 0.6946}{0.9781} \).
• Finally, calculate and round \( c \) to get about 18.3.

By understanding and applying the Law of Sines, we can effectively "solve" triangle ABC and find all required elements.