Problem 18 - Precalculus with Calculus Previews

Question

Use the Law of Sines to solve the triangle. $$ \alpha=140^{\circ}, \gamma=20^{\circ}, c=12 $$

Step-by-Step Solution

Answer

The side lengths are: \(a \approx 22.56\), \(b = 12\), \(c = 12\).

Step 1: Calculate Angle Beta

To find angle \( \beta \), use the fact that the sum of angles in a triangle is \( 180^\circ \). Thus, \( \beta = 180^\circ - \alpha - \gamma = 180^\circ - 140^\circ - 20^\circ = 20^\circ \).

Step 2: Apply the Law of Sines for Side a

According to the Law of Sines, \( \frac{a}{\sin \alpha} = \frac{c}{\sin \gamma} \). Substitute the known values: \( \frac{a}{\sin 140^\circ} = \frac{12}{\sin 20^\circ} \).

Step 3: Solve for Side a

Calculate \( \sin 140^\circ \approx 0.643 \) and \( \sin 20^\circ \approx 0.342 \). Solve the equation: \( a = \frac{12 \times 0.643}{0.342} \approx 22.56 \).

Step 4: Apply the Law of Sines for Side b

Use the relation \( \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \). Substitute the known values: \( \frac{b}{\sin 20^\circ} = \frac{12}{\sin 20^\circ} \).

Step 5: Solve for Side b

Since \( \sin 20^\circ \) cancels out on both sides, \( b = 12 \).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles. It's particularly useful when the shapes you are dealing with are not simple right triangles. In trigonometry, each angle of the triangle has a corresponding side opposite it, and these relationships are governed by trigonometric functions like sine, cosine, and tangent.
One of the powerful tools in trigonometry is the Law of Sines, which is particularly helpful in solving triangles that are not right-angled. This law states that for any triangle with angles \( \alpha, \beta, \gamma \) and respective opposite sides \( a, b, c \), the ratio of the length of a side to the sine of its opposite angle is constant. In formula terms, it is expressed as:
\[ \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \]
This formula allows you to relate the angles and sides of any given triangle, making it indispensable for solving a broad range of triangle problems.

Solving Triangles

Solving triangles involves finding missing side lengths and angle measures in a triangle, using given values and mathematical relationships. When you have some information about a triangle, such as angles and one side length, you can use the Law of Sines to determine the unknowns. In the given exercise, you have two angles, \( \alpha = 140^{\circ} \) and \( \gamma = 20^{\circ} \), along with side \( c = 12 \).
To solve the triangle, you can follow these basic steps:

• First, use the angle sum property to find the missing angle \( \beta \).
• Next, apply the Law of Sines to determine the missing side lengths \( a \) and \( b \).

For angle \( \beta \), use the formula **\( 180^{\circ} - \alpha - \gamma \)** to find that \( \beta = 20^{\circ} \).
Using the Law of Sines and substituting known values, like \( \sin 140^{\circ} \approx 0.643 \) and \( \sin 20^{\circ} \approx 0.342 \), you can solve for the side lengths. Solving triangles becomes a straightforward process once you understand how these mathematical tools interact.

Angle Sum Property

The angle sum property is a fundamental concept in geometry and trigonometry. It states that the sum of the internal angles of a triangle is always \( 180^{\circ} \). This property is especially important when dealing with the Law of Sines, as it allows you to find unknown angles given the other two. For instance, in a triangle where you know two angles, say \( \alpha \) and \( \gamma \), you can easily find the third angle \( \beta \) using:
\[ \beta = 180^{\circ} - \alpha - \gamma \]
In the original exercise, knowing \( \alpha = 140^{\circ} \) and \( \gamma = 20^{\circ} \), we apply this property to find \( \beta \). It's as simple as subtracting the sum of the known angles from \( 180^{\circ} \), resulting in \( \beta = 20^{\circ} \).
The angle sum property is not only crucial in trigonometry but also forms the basis for understanding more complex geometric shapes. It's one of the basic building blocks that you'll repeatedly encounter as you progress in mathematics.