Problem 7 - Trigonometry

Question

If \(B=120^{\circ}, C=20^{\circ}\), and \(c=28\) inches, find \(b\).

Step-by-Step Solution

Answer

The length of side \( b \) is approximately 70.53 inches.

Step 1: Understanding Given Values

We are given two angles of a triangle, namely \( B = 120^{\circ} \) and \( C = 20^{\circ} \), and one side \( c = 28 \) inches, opposite to angle \( C \). We need to find the length of side \( b \), which is opposite to angle \( B \).

Step 2: Calculate the Third Angle

Use the sum of angles in a triangle to find the third angle \( A \). The sum of angles in a triangle is \( 180^{\circ} \). Hence,\( A = 180^{\circ} - B - C \). Substitute the known values: \( A = 180^{\circ} - 120^{\circ} - 20^{\circ} = 40^{\circ} \).

Step 3: Apply the Law of Sines

The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We only need to use parts of this formula involving \( b \) and \( c \): \( \frac{b}{\sin B} = \frac{c}{\sin C} \).

Step 4: Solve for \( b \)

Rearrange the equation to solve for \( b \): \( b = \frac{c \cdot \sin B}{\sin C} \). Substitute the known values: \( b = \frac{28 \cdot \sin(120^{\circ})}{\sin(20^{\circ})} \).

Step 5: Calculate Sine Values

Compute \( \sin(120^{\circ}) \) and \( \sin(20^{\circ}) \). \( \sin(120^{\circ}) = \sin(180^{\circ} - 60^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \). Use a calculator for \( \sin(20^{\circ}) \): it is approximately \( 0.342 \).

Step 6: Final Calculation

Substitute the sine values into the equation for \( b \): \( b = \frac{28 \cdot \frac{\sqrt{3}}{2}}{0.342} \). Simplify the expression by calculating the numerical approximation to find \( b \) approximately equals \( 70.53 \) inches.

Key Concepts

Understanding Triangle Angles

In a triangle, the angles have a special relationship, which is essential for solving numerous geometry problems. The sum of the interior angles of any triangle is always 180 degrees. This fact is crucial when we need to determine a missing angle if two angles are known.

• For the given problem, angles B and C are 120° and 20° respectively.
• To find the missing angle A, you subtract the sum of the known angles from 180°: \[ A = 180^{\circ} - B - C = 180^{\circ} - 120^{\circ} - 20^{\circ} = 40^{\circ} \]

This property allows us to complete our understanding of how the angles relate within the triangle, hence aiding in further calculations, such as solving for unknown sides.

Solving Triangles Using the Law of Sines

Solving triangles often entails finding the unknown angles or sides using known dimensions. One effective method for solving oblique triangles, which are non-right triangles, is the Law of Sines. This law is expressed as:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]In our scenario, the focus is on finding side \( b \), with angle \( B \) already known. We can set up the equation:

• \( \frac{b}{\sin B} = \frac{c}{\sin C} \)
• Substitute the known values and rearrange to solve for \( b \):\[ b = \frac{c \cdot \sin B}{\sin C} \]

This approach simplifies tackling problems involving non-right triangles by creating a proportion based on the angles and sides, leading us towards finding the desired measurements.

Deep Dive into the Sine Function

The sine function is part of trigonometry and plays a pivotal role in solving various geometric problems, including those involving triangles. It represents the ratio of the opposite side to the hypotenuse in a right triangle. However, it is also applicable to non-right triangles via the Law of Sines.

• The sine function helps us link angles and sides effectively, thus calculating unknown elements of a triangle.
• For instance, calculate the sine values for given angles, like in our problem:
  • \( \sin(120^{\circ}) = \sin(180^{\circ} - 60^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \)
  • Approximating \( \sin(20^{\circ}) \) with a calculator gives about 0.342.

These values are crucial for solving problems using the Law of Sines, enabling us to make necessary calculations for determining unknown segments, such as side \( b \) in this scenario.