Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Solve \(\triangle A B C\). $$\alpha=103.45^{\circ}, \quad \gamma=27.19^{\circ}, \quad b=38.84$$

Step-by-Step Solution

Answer

\(\beta = 49.36^{\circ}\), \(a \approx 48.55\), \(c \approx 23.69\)

Step 1: Identify Remaining Angle

The sum of the angles in a triangle is always 180 degrees. Therefore, to find the missing angle \(\beta\) in triangle \(\triangle ABC\), use the formula: \(\alpha + \beta + \gamma = 180^{\circ}\).

Step 2: Calculate the Missing Angle

Substitute the given values of \(\alpha\) and \(\gamma\) into the angle sum equation: \(103.45^{\circ} + \beta + 27.19^{\circ} = 180^{\circ}\), and solve for \(\beta\).

Step 3: Solve for Missing Angle Value

Calculate \(\beta\) using the previous equation: \(\beta = 180^{\circ} - 103.45^{\circ} - 27.19^{\circ}\). This gives \(\beta = 49.36^{\circ}\).

Step 4: Apply Law of Sines

To find the missing sides \(a\) and \(c\), use the Law of Sines: \(\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}\). First, solve for side \(a\).

Step 5: Calculate Side a

Using \(b = 38.84\) and \(\beta = 49.36^{\circ}\), solve for \(a\) using \(\frac{a}{\sin 103.45^{\circ}} = \frac{38.84}{\sin 49.36^{\circ}}\). Rearrange to find \(a = \frac{38.84 \cdot \sin 103.45^{\circ}}{\sin 49.36^{\circ}}\).

Step 6: Compute Side a

Calculate \(a\) using a calculator: \(a \approx 48.55\).

Step 7: Solve for Side c

Similarly, use the Law of Sines to find \(c\): \(\frac{c}{\sin 27.19^{\circ}} = \frac{38.84}{\sin 49.36^{\circ}}\), rearrange to find \(c = \frac{38.84 \cdot \sin 27.19^{\circ}}{\sin 49.36^{\circ}}\).

Step 8: Compute Side c

Calculate \(c\) using a calculator: \(c \approx 23.69\).

Step 9: Summary

The triangle \(\triangle ABC\) has \(\beta = 49.36^{\circ}\), \(a \approx 48.55\), and \(c \approx 23.69\).

Key Concepts

Angle Sum Property

Every triangle, regardless of its shape or type, follows a fundamental rule known as the angle sum property. This property states that the sum of the interior angles of a triangle is always 180 degrees. This is a crucial concept when solving triangles because knowing any two angles allows you to find the third.

For instance, in triangle \( \triangle ABC \), if you know two angles like \( \alpha = 103.45^{\circ} \) and \( \gamma = 27.19^{\circ} \), you can calculate the missing angle \( \beta \) using:

• \( \alpha + \beta + \gamma = 180^{\circ} \)
• Substitute the known values to find the unknown: \( 103.45^{\circ} + \beta + 27.19^{\circ} = 180^{\circ} \)
• Solve for \( \beta \): \( \beta = 180^{\circ} - 103.45^{\circ} - 27.19^{\circ} \)

This method is straightforward yet powerful in determining any missing angle in a triangle.

Law of Sines

The Law of Sines is a fundamental tool used in trigonometry to solve triangles. It relates the sides of a triangle to the sines of their opposite angles. Knowing this property lets you solve for unknown sides and angles with ease.

The formula is written as:

• \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \)

Using the Law of Sines becomes especially useful when you have either:

• Two angles and a side (AAS or ASA condition)
• Or two sides and an opposite angle (SSA condition)

If we know \( b \) and \( \beta \), as well as another angle, such as \( \alpha \), we can find unknown side \( a \) by rearranging the equation:

• \( a = \frac{b \cdot \sin \alpha}{\sin \beta} \)

This helps in calculating unknown sides like \( a \) and \( c \) efficiently.

Triangle Angles

Understanding triangle angles is vital in solving and working with triangles effectively. Each angle is directly related to the triangle's sides and overall properties. Remembering the

• Sum of angles is always \( 180^{\circ} \)
• Use of supplementary and complementary angle concepts

can significantly aid in solving problems.

For interesting cases where you need to compare angles, the larger the angle, the longer the side opposite it. Conversely, smaller angles will have shorter opposite sides. In \( \triangle ABC \), knowing \( \alpha = 103.45^{\circ} \), \( \gamma = 27.19^{\circ} \), and \( \beta = 49.36^{\circ} \) demonstrates this nicely as \( \alpha \) is the largest, making side \( a \) the longest.

Triangle Sides

The lengths of a triangle's sides carry information about both the shape and the type of the triangle. Triangle side problems usually involve either finding unknown side lengths or proving the triangle type, like isosceles or scalene.

When solving for triangle sides, such as in \( \triangle ABC \), we frequently use:

• The Law of Sines (as illustrated earlier)
• Basic trigonometric identities if applicable

A thorough understanding of how the sides relate through the angles and using equations like the Law of Sines ensures that each side's length is correctly calculated. In our example, knowing side \( b \) enables us to solve for the remaining sides \( a \) and \( c \) using simple

• Trigonometric calculations

These techniques solve the triangle completely, allowing you to work with all its properties confidently.