Problem 11 - Trigonometry

Question

If \(A=50^{\circ}, B=60^{\circ}\), and \(a=36 \mathrm{~km}\), find \(C\) and then find \(c\).

Step-by-Step Solution

Answer

Angle C is \(70^\circ\), and side c is approximately 44.3 km.

Step 1: Identify Given Angles and Use Triangle Sum

Given angles are \(A = 50^\circ\) and \(B = 60^\circ\). Since the sum of angles in a triangle is \(180^\circ\), use the formula \(C = 180^\circ - A - B\) to find \(C\).

Step 2: Calculate Angle C

Plug the values of \(A\) and \(B\) into the formula: \[C = 180^\circ - 50^\circ - 60^\circ = 70^\circ.\] So, angle \(C\) is \(70^\circ\).

Step 3: Apply the Sine Rule to Find Side c

The Sine Rule states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Use \( \frac{a}{\sin A} = \frac{c}{\sin C} \) to find \(c\).

Step 4: Substitute Known Values into the Sine Rule

Use \(a = 36\), \(A = 50^\circ\), and \(C = 70^\circ\). Substitute them into the Sine Rule: \[ \frac{36}{\sin 50^\circ} = \frac{c}{\sin 70^\circ}. \]

Step 5: Solve for Side c

Rearrange the equation to find \(c\): \[ c = \frac{36 \cdot \sin 70^\circ}{\sin 50^\circ}. \] Calculate using a calculator: \[ c \approx \frac{36 \cdot 0.9397}{0.7660} \approx 44.3 \text{ km}. \]

Key Concepts

Triangle Sum Theorem

A fundamental concept in triangle geometry is the Triangle Sum Theorem, which states that the sum of the interior angles of a triangle is always equal to 180 degrees. This theorem is very helpful because it allows us to find a missing angle when we know the other two angles.

For example, if you know angles \(A\) and \(B\) in a triangle, you can find angle \(C\) by using this simple formula:

• \(C = 180^{\circ} - A - B\)

In this exercise, angles \(A\) and \(B\) are given as \(50^{\circ}\) and \(60^{\circ}\) respectively. By substituting these into the formula, we calculate angle \(C\) as \(70^{\circ}\). Knowing how to apply this theorem helps us handle various triangle-related problems confidently.

Angle Calculation

Calculating the angles in a triangle requires careful substitution and addition based on known information. In the provided example, we knew angles \(A\) and \(B\). This information enabled us to find angle \(C\) effectively.

Here's a stepwise approach:

• Ensure all known angles are accounted for correctly. Here, \(A = 50^{\circ}\) and \(B = 60^{\circ}\).
• Use the Triangle Sum Theorem to solve for the missing angle \(C\), calculating as follows: \(180^{\circ} - 50^{\circ} - 60^{\circ} = 70^{\circ}\).
• Finally, confirm that the computed angle makes sense (e.g., not exceeding 180 for the sum of angles).

This keeps us on track and guarantees that angles remain feasible and logical in the context of triangles.

Side Calculation

Finding the missing side of a triangle often involves applying the Sine Rule, especially in non-right triangles. The Sine Rule relates the sides of a triangle to the sines of their opposite angles and is given by:

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

This rule is particularly useful when we know at least one side and its opposite angle, enabling us to find other sides efficiently.
In our scenario, we applied the Sine Rule to calculate side \(c\), knowing \(a = 36 \text{ km}\), \(A = 50^{\circ}\), and \(C = 70^{\circ}\). By setting up the equation \(\frac{36}{\sin 50^{\circ}} = \frac{c}{\sin 70^{\circ}}\), we solved for \(c\) by rearranging and plugging in the sine values using a calculator:

• \(c = \frac{36 \times \sin 70^{\circ}}{\sin 50^{\circ}}\)
• This gives approximately \(44.3 \text{ km}\).

Using the Sine Rule bridges the gap between angles and sides, thus facilitating the complete understanding of triangle properties.