Problem 13 - Trigonometry

Question

If \(A=52^{\circ}, B=48^{\circ}\), and \(c=14 \mathrm{~cm}\), find \(C\) and then find \(a\).

Step-by-Step Solution

Answer

Angle C is 80°, and side \(a\) is approximately 11.20 cm.

Step 1: Understand the Triangle

A triangle has angles A, B, and C. The sum of internal angles in a triangle is always 180°. We can use this fact to find angle C since angles A and B are given.

Step 2: Calculate Angle C

Given that angle A is 52° and angle B is 48°, we find angle C using the equation: \[ C = 180° - (A + B) \]Substituting the given values, \[ C = 180° - (52° + 48°) \]\[ C = 180° - 100° \]\[ C = 80° \]

Step 3: Apply the Law of Sines

To find side \(a\), we use the Law of Sines:\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]We are given \(c = 14\, \text{cm}\), \(A = 52^{\circ}\), and we found \(C = 80^{\circ}\). Calculate \(\sin A\) and \(\sin C\):\[ \sin 52^{\circ} \approx 0.7880 \]\[ \sin 80^{\circ} \approx 0.9848 \]

Step 4: Solve for Side a

Using the equation from the Law of Sines:\[ a = \frac{c \cdot \sin A}{\sin C} \]Substitute the known values:\[ a = \frac{14 \cdot 0.7880}{0.9848} \]\[ a \approx \frac{11.032}{0.9848} \]\[ a \approx 11.20 \]

Key Concepts

Law of Sines

The Law of Sines provides a powerful tool to solve non-right triangles. It's particularly useful when you know either:

• Two angles and one side (AAS or ASA)
• Two sides and a non-included angle (SSA)

The Law of Sines states that the ratios of the lengths of sides to the sines of their opposite angles are equal. Mathematically, it can be expressed as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]In our triangle with sides labeled as \(a, b,\) and \(c\) and angles \(A, B,\) and \(C\), let’s imagine that you need to find a missing side or angle. Suppose you have computed angle \(C\) to be \(80^{\circ}\), and side \(c\) is \(14 \text{ cm}\). This is a classic AAS scenario. You can swap in the known angle-side pairs to solve for the unknown side \(a\), utilizing sine values from trigonometric tables or a calculator, just like we did.

Triangle Properties

Triangles are simple yet incredibly important shapes in geometry. They have three sides and three angles. One of the most crucial properties of a triangle is that the sum of its internal angles always equals \(180^{\circ}\).
This fundamental rule was pivotal in our exercise:

• Known angles \(A = 52^{\circ}\) and \(B = 48^{\circ}\) enabled the calculation of angle \(C\).

It doesn’t matter how the triangle looks physically; this rule holds strong for all.
Additionally, there are different types of triangles based on sides and angles. When two angles are provided along with a non-included side (AAS configuration), we can easily find the unknowns by using properties like the angle sum and Law of Sines. This helps in solving challenges in fields like physics, engineering, and architecture.

Angle Calculation

Calculating an unknown angle in a triangle uses one straightforward rule: the angles sum to \(180^{\circ}\). When some angles are known, you subtract their sum from \(180^{\circ}\) to find the missing angle.
Here’s how we calculated angle \(C\):

• Start with known \(A = 52^{\circ}\) and \(B = 48^{\circ}\). Notice they together account for \(100^{\circ}\).
• Subtract \(100^{\circ}\) from \(180^{\circ}\) to find \(C\): \(80^{\circ}\).

Unlike more complex equations, this makes solving for angles quick and straightforward. Armed with this and the Law of Sines, you can tackle most non-right triangle problems head-on. Keep in mind that these calculations often rely on finding precise sine values using a calculator or table for accurate results.