Problem 8 - 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. $$ \alpha=42^{\circ}, a=17, b=23.5 $$

Step-by-Step Solution

Answer

\( \beta \approx 68.47^{\circ}, \gamma \approx 69.53^{\circ}, c \approx 24.6 \).

Step 1: Identify the Known Elements

We are given \( \alpha = 42^{\circ} \), \( a = 17 \), and \( b = 23.5 \). We need to find \( \beta \), \( \gamma \), and \( c \). We will use the Law of Sines and the fact that angles in a triangle sum to 180°.

Step 2: Apply the Law of Sines to Find \( \beta \)

The Law of Sines states \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \). This can be rearranged to \( \sin(\beta) = \frac{b \cdot \sin(\alpha)}{a} \). Substituting in the known values gives \( \sin(\beta) = \frac{23.5 \cdot \sin(42^{\circ})}{17} \), which approximates to \( \sin(\beta) \approx 0.9345 \). Using the inverse sine function, \( \beta \approx 68.47^{\circ} \).

Step 3: Calculate \( \gamma \) Using Angle Sum Property

In any triangle, the sum of angles is \(180^{\circ}\). We know \( \alpha \) and \( \beta \), so we can find \( \gamma \) using \( \gamma = 180^{\circ} - \alpha - \beta = 180^{\circ} - 42^{\circ} - 68.47^{\circ} \approx 69.53^{\circ} \).

Step 4: Use the Law of Sines to Solve for \( c \)

Now that we know all the angles, use the Law of Sines to find \( c \): \( \frac{c}{\sin(\gamma)} = \frac{a}{\sin(\alpha)} \). Rearranging gives \( c = \frac{a \cdot \sin(\gamma)}{\sin(\alpha)} \). Substituting the known values results in \( c \approx \frac{17 \cdot \sin(69.53^{\circ})}{\sin(42^{\circ})} \), yielding \( c \approx 24.6 \).

Key Concepts

Understanding Angle-Side Pairs

When working with triangles, angle-side pairs are crucial components to keep in mind. An angle-side pair consists of an angle in a triangle and its opposite side. These pairs are important because they allow us to apply the Law of Sines effectively. For example, in a triangle with angles \(\alpha\), \(\beta\), and \(\gamma\), and corresponding sides \(a\), \(b\), and \(c\), \(\alpha\) and \(a\) form one angle-side pair. These pairs help us set up equations that relate the sides and angles of the triangle.

Understanding these relationships is key when solving triangles, as knowing one angle-side pair can often help us find other unknowns in the problem with tools like the Law of Sines. With angle-side pairs, you have the foundational building blocks for solving the rest of the triangle.

The Angle Sum Property Explained

Triangles inherently follow several geometric properties, one of which is the angle sum property. This property states that the sum of all interior angles in a triangle equals 180 degrees.

This is a fundamental concept that can be used to find a missing angle when the other two angles are known. For instance, if you know angles \(\alpha\) and \(\beta\) in a triangle, you can always calculate angle \(\gamma\) using the formula \(\gamma = 180^{\circ} - \alpha - \beta\).

The angle sum property is straightforward, yet it’s one of the most powerful tools when solving triangles because it guarantees that we can always calculate the third angle, provided we know the other two angles.

Solving Triangles Using the Law of Sines

When we talk about solving triangles, our goal is to find all the unknown angles and sides. The Law of Sines is particularly useful for this purpose when we have at least one angle-side pair.

The Law of Sines is expressed as: \(\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}\). This relationship is valid in any triangle, allowing us to find unknown parts given enough initial information. This is especially helpful in non-right triangles, where standard trigonometric functions like sine and cosine are less straightforward to apply directly.

Using the Law of Sines, you can easily resolve angles or sides when you possess the necessary initial data. For example, given an angle-side pair \((\alpha, a)\), and another length \(b\), you can calculate \(\beta\) and ultimately solve for all parts of the triangle.

Utilizing the Inverse Sine Function

To find an angle from a known sine value, we use the inverse sine function. This is crucial when solving for angles like \(\beta\) once you have determined \(\sin(\beta)\) using the Law of Sines.

The inverse sine function, written as \(\sin^{-1}\) or \(\arcsin\), helps convert a sine ratio back into an angle measurement. For example, if \(\sin(\beta) \approx 0.9345\), then \(\beta \approx \sin^{-1}(0.9345)\).

It’s important to remember that output from the inverse sine function is usually in the range \(-90^{\circ}\) to \(90^{\circ}\), so ensure your angle makes sense within the context of the problem. This function is pivotal in the process of angle determination to achieve full resolution of the triangle's dimensions.