Problem 23 - Trigonometry

Question

Find all solutions to each of the following triangles: \(C=26.8^{\circ}, c=36.8 \mathrm{~km}, b=36.8 \mathrm{~km}\)

Step-by-Step Solution

Answer

The triangle is isosceles with angles \( A = 126.4^{\circ} \), \( B = 26.8^{\circ} \), and all sides \( b = c = 36.8 \text{ km} \).

Step 1: Analyze Given Information

The triangle given has \( C = 26.8^{\circ} \), \( c = 36.8 \text{ km} \), and \( b = 36.8 \text{ km} \). Since \( b = c \), it indicates an isosceles triangle with \( C \) as the angle between the equal sides.

Step 2: Use the Sine Rule to Determine Angle B

Using the sine rule, which states \( \frac{b}{\sin B} = \frac{c}{\sin C} \), we have \( \frac{36.8}{\sin B} = \frac{36.8}{\sin 26.8^{\circ}} \). Solving for \( \sin B \), we find \( \sin B = \sin 26.8^{\circ} \). Thus, \( B = 26.8^{\circ} \).

Step 3: Recognize Triangle Type

With \( B = C = 26.8^{\circ} \), we realize the triangle is also isosceles in terms of angles, making it an isosceles triangle with two equal angles and two equal sides.

Step 4: Find Angle A

Using the angle sum property of triangles, angle \( A \) is calculated as \( A = 180^{\circ} - B - C = 180^{\circ} - 26.8^{\circ} - 26.8^{\circ} = 126.4^{\circ} \).

Step 5: Verify the Triangle Solution

Check if the sum of angles \( A + B + C = 180^{\circ} \) holds true. Calculating: \( 126.4^{\circ} + 26.8^{\circ} + 26.8^{\circ} = 180^{\circ} \), confirming the solution is correct.

Key Concepts

Isosceles Triangles

Isosceles triangles are a special type of triangle where two sides are equal in length, which also results in two angles being equal. This unique symmetry gives isosceles triangles a few interesting properties that can simplify solving them.

• Understanding the triangle begins with identifying the equal sides. In our problem, sides \( b \) and \( c \) are both \( 36.8 \text{ km} \).
• When two sides are equal, the angles opposite these sides are equal. Hence, angles \( B \) and \( C \) are both \( 26.8^{\circ} \) in this isosceles triangle.

Recognizing an isosceles triangle can quickly tell you a lot about its angles and sides without further calculation. It provides a shortcut to understanding the relationship among its components.

Sine Rule

The sine rule is a vital tool for solving triangles when you know elements like sides and angles. The sine rule relates the sides of a triangle to the sines of its angles and can be incredibly useful in cases such as this. The formula is:

• \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

In the scenario of our exercise, the problem setup already suggests using this rule because we have a side-angle-side (SAS) scenario with known angle \( C \) and its opposite side \( c \). This allows us to find angle \( B \) using:

• \[ \frac{b}{\sin B} = \frac{c}{\sin C} \]
• In this specific case, \( \sin B = \sin 26.8^{\circ} \), which directly results in \( B = 26.8^{\circ} \).

The sine rule is an excellent choice for such problems because it provides an immediate relationship between the sides and angles across the triangle.

Angle Sum Property

The sum of angles in any triangle is always \( 180^{\circ} \). This fundamental property is very helpful for solving triangles, especially when two angles are known, as it makes finding the third straightforward.

• Using the angle sum property, if we know \( B \) and \( C \), we can find \( A \).
• We apply the formula \( A = 180^{\circ} - B - C \).
• In our specific exercise, this calculation becomes \( A = 180^{\circ} - 26.8^{\circ} - 26.8^{\circ} = 126.4^{\circ} \).

Thus, using the angle sum property provides a straightforward way to complete the triangle once two angles and one side are known, ensuring the solution is consistent and correct.