Problem 34 - A Graphical Approach to Precalculus with Limits

Question

Solve each triangle. \(B=168.2^{\circ}, a=15.1\) centimeters, \(c=19.2\) centimeters

Step-by-Step Solution

Answer

The angles are \(A \approx 9.2^{\circ}\), \(B = 168.2^{\circ}\), \(C = 11.8^{\circ}\) and the sides are \(a = 15.1\) cm, \(b \approx 0.58\) cm, \(c = 19.2\) cm.

Step 1: Identify the known values

We are given that angle \(B = 168.2^{\circ}\), side \(a = 15.1\) cm, and side \(c = 19.2\) cm. We need to find angle \(A\), angle \(C\), and side \(b\).

Step 2: Use the Law of Sines

The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). First, calculate \(\sin B\) using the given angle \(B = 168.2^{\circ}\).

Step 3: Calculate \(\sin B\)

Calculate \(\sin B = \sin(168.2^{\circ})\). Use a calculator to find this value: \(\sin 168.2^{\circ} \approx 0.1736\).

Step 4: Find angle \(A\) using the Law of Sines

Using the Law of Sines, \(\frac{a}{\sin A} = \frac{c}{\sin C}\), we rearrange it to find \(\sin A = \frac{a \cdot \sin C}{c}\). However, first we must find \(C\) before \(A\).

Step 5: Find angle \(C\) using angles in a triangle

The sum of angles in any triangle is \(180^{\circ}\). Therefore, \(C = 180^{\circ} - B = 180^{\circ} - 168.2^{\circ} = 11.8^{\circ}\).

Step 6: Use the Law of Sines to find \(A\)

With \(C = 11.8^{\circ}\), use \(\frac{a}{\sin A} = \frac{c}{\sin C}\) to find \(\sin A = \frac{15.1 \cdot \sin 11.8^{\circ}}{19.2}\). Calculate \(\sin 11.8^{\circ} \approx 0.2041\). Thus, \(\sin A \approx 0.1602\).

Step 7: Solve for angle \(A\)

Use \(\sin^{-1}\) to find \(A = \sin^{-1}(0.1602)\). Use a calculator to find \(A \approx 9.2^{\circ}\).

Step 8: Use the Law of Sines to find side \(b\)

Now, using \(\frac{b}{\sin B} = \frac{a}{\sin A}\), solve for \(b\): \(b = \frac{15.1 \cdot \sin 168.2^{\circ}}{\sin 9.2^{\circ}}\). Substitute in the values and find \(b \approx 0.58\) cm.

Key Concepts

Law of Sines

The **Law of Sines** is a fundamental principle in trigonometry to solve triangles, especially non-right triangles. It establishes a relationship between the sides of a triangle and their opposite angles. This law is expressed as:

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

Here, \(a, b,\) and \(c\) are the sides of the triangle, and \(A, B,\) and \(C\) are the angles opposite those sides.

The **Law of Sines** is used when we know:

• Two angles and one side (AAS or ASA case)
• Two sides and a non-included angle (SSA case)

In our problem, the two given sides \(a\) and \(c\), and angle \(B\) allow us to use this law to solve for the missing parts of the triangle. By calculating the sine of angle \(B\) and using the known sides, we can find remaining angles and the third side.

Triangle Angles

Understanding the **Triangle Angles** principle is crucial for solving any triangle problem. In any triangle, the sum of all interior angles is always \(180^{\circ}\). This is known as the Triangle Sum Theorem and can be a handy way to find unknown angles.

Given that angle \(B = 168.2^{\circ}\) in the problem, the formula is applied as:

• \(C = 180^{\circ} - B = 180^{\circ} - 168.2^{\circ}\)

This results in \(C = 11.8^{\circ}\).

Knowing two of the angles directly allows us to find the third angle \(A\) by subtraction when needed, reinforcing the interconnected nature of triangle angles.

Trigonometric Functions

**Trigonometric Functions** are essential mathematical tools that establish connections between angles and ratios of sides in a triangle. The three primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). For our problem, knowing sine was particularly useful.

Sine helps relate the angle of a triangle to its opposite side length. For angle \(B\) given in our problem:

• \(\sin B = \sin(168.2^{\circ}) \approx 0.1736\)

This calculation is key for applying the Law of Sines to find unknown side lengths or angles.

Moreover, the inverse sine function, or \(\sin^{-1}\), allows us to retrieve angle values from known sine ratios, such as when we calculated \(A = \sin^{-1}(0.1602) \approx 9.2^{\circ}\). These trigonometric functions help bridge the operations required to solve any triangle effectively.