Problem 12 - College Trigonometry

Question

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \gamma=53^{\circ}, \alpha=53^{\circ}, c=28.01 $$

Step-by-Step Solution

Answer

The missing angle \(\beta\) is \(74^\circ\), and the side lengths are \(a = 28.01\) and \(b \approx 33.71\).

Step 1: Recognize Triangle Properties

In any triangle, the sum of all angles is always 180 degrees. Since you have been given two angles, \(\gamma = 53^\circ\) and \(\alpha = 53^\circ\), you can calculate the third angle, \(\beta\).

Step 2: Calculate the Missing Angle \(\beta\)

To find \(\beta\), use the formula for the sum of angles in a triangle: \(\alpha + \beta + \gamma = 180^\circ\). Substitute the known angles into the formula: \(53^\circ + 53^\circ + \beta = 180^\circ\). Solve for \(\beta\) to find \(\beta = 180^\circ - 106^\circ = 74^\circ\).

Step 3: Use the Law of Sines to Find \(a\) and \(b\)

The Law of Sines states \(\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}\). Start by finding \(a\). Substitute the known values: \(\frac{a}{\sin 53^\circ} = \frac{28.01}{\sin 53^\circ}\), which simplifies to \(a = 28.01\). Now solve for \(b\) using \(\frac{b}{\sin 74^\circ} = \frac{28.01}{\sin 53^\circ}\). Calculate \(b = \frac{28.01 \cdot \sin 74^\circ}{\sin 53^\circ}\).

Step 4: Calculate \(b\)

Substitute the known values into the equation from the Law of Sines: \(b = \frac{28.01 \cdot \sin 74^\circ}{\sin 53^\circ}\). Calculate the sines: \(\sin 74^\circ \approx 0.961\) and \(\sin 53^\circ \approx 0.799\). Now plug in these values to get \(b \approx \frac{28.01 \cdot 0.961}{0.799} \approx 33.71\).

Key Concepts

Sum of Angles in a Triangle

In any triangle, a very important property to remember is that the sum of its internal angles is always 180 degrees. It doesn't matter what type of triangle you are dealing with—this rule is universal. This property helps us find unknown angles when at least two angles in the triangle are known.
For example, if you're given two angles, \(\alpha\) and \(\gamma\), you can find the third angle, \(\beta\), by subtracting the sum of \(\alpha\) and \(\gamma\) from 180 degrees. In the exercise, \(53^\circ + 53^\circ\), gives a sum of 106 degrees, so \(\beta = 180^\circ - 106^\circ = 74^\circ\).
This subset of knowledge is fundamental in various applications of geometry and helps in both practical and theoretical problem-solving.

Law of Sines

The Law of Sines is an incredibly useful tool when working with triangles, particularly non-right triangles. It relates the lengths of the sides of a triangle to the sines of its angles. The law states: \(\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}\).
Essentially, for any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. It's particularly handy when we know:

• Two angles and one side
• Two sides and a non-included angle

In our exercise, \(\gamma\) and \(\alpha\) are both \(53^\circ\), and side \(c\) is given as 28.01. Using the Law of Sines, we determine that side \(a\), opposite to \(\alpha\), equals side \(c\) because \(\alpha = \gamma\).
For the last part, to find \(b\), we continue using the Law of Sines and substitute: \(\frac{b}{\sin 74^\circ} = \frac{28.01}{\sin 53^\circ}\). This yields \(b \approx 33.71\).
Understanding and applying the Law of Sines allows for easier navigation through problems involving triangles.

Solving Triangles

Solving triangles involves finding unknown angles or sides given some initial information. This exercise combined concepts such as the sum of angles in a triangle and the Law of Sines to efficiently solve the triangle.
Here's the general approach to solving a triangle:

• First, determine if you know enough to solve the triangle. You usually need at least one side and enough angles or another side to get started.
• If only angles are missing, use the sum of angles. If sides are involved, consider the Law of Sines or Cosines.
• Lastly, check your results. Make sure your calculated angles add to 180 degrees, and that any side inequalities hold true (e.g., the largest side is opposite the largest angle).

In our specific case, after finding the missing angle using the sum of angles, knowing side \(c\), and employing the Law of Sines, we efficiently calculated sides \(a\) and \(b\). This step-by-step method ensures each measurement and calculation makes logical sense and fits perfectly within the triangle's properties.