Problem 15 - College Trigonometry

Question

Solve for the remaining side(s) and angle(s), if possible, using any appropriate technique. $$ \alpha=42^{\circ}, b=117, c=88 $$

Step-by-Step Solution

Answer

The unknown side is \( a \approx 88.06 \), and the angles are \( \beta \approx 62.75^{\circ} \) and \( \gamma \approx 75.25^{\circ} \).

Step 1: Use the Law of Sines

The Law of Sines states that \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \). We can use this to find angle \( \beta \). Substitute: \[ \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \] \[ \frac{117}{\sin \beta} = \frac{88}{\sin 42^{\circ}} \] First, calculate \( \sin 42^{\circ} \). Use a calculator to find that \( \sin 42^{\circ} \approx 0.6691 \). Substituting this value, we get: \[ \frac{117}{\sin \beta} = \frac{88}{0.6691} \] \[ \sin \beta \approx \frac{117 \cdot 0.6691}{88} \approx 0.889 \]

Step 2: Calculate Angle \( \beta \)

With \( \sin \beta \approx 0.889 \), use the inverse sine function to determine \( \beta \). Using a calculator, \( \beta \approx \sin^{-1}(0.889) \approx 62.75^{\circ} \).

Step 3: Calculate Angle \( \gamma \)

The sum of angles in a triangle is \( 180^{\circ} \). Hence, we can find \( \gamma \) using the equation: \[ \gamma = 180^{\circ} - \alpha - \beta \] \[ \gamma = 180^{\circ} - 42^{\circ} - 62.75^{\circ} \approx 75.25^{\circ} \]

Step 4: Calculate Side \( a \) Using the Law of Sines Again

Now that we have all angles, we can use the Law of Sines to find side \( a \). \[ \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} \rightarrow a = \frac{b \cdot \sin \alpha}{\sin \beta} \] Substituting the known values: \[ a = \frac{117 \cdot \sin 42^{\circ}}{\sin 62.75^{\circ}} \approx \frac{117 \cdot 0.6691}{0.889} \approx 88.06 \]

Key Concepts

Triangle Solving

When solving a triangle, we aim to find missing angles or sides using given values. In the given problem, we're provided with two side lengths, and one angle. The first step is often to use trigonometric principles or specific theorems to work towards a solution. Drawing the triangle could help visualize the known and unknown components, which makes tackling the problem more manageable.
One of the indispensable tools in triangle solving is the Law of Sines. This law relates the sides and angles of a triangle. In a triangle with angles \(\alpha, \beta, \gamma\) and corresponding opposite sides \(a, b, c\), respectively, the Law of Sines is expressed as:

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

We'll use this to find missing angles and sides. In triangle solving, ensure you have a solid grasp of trigonometric functions and identities to efficiently apply the theorem.

Angle Calculation

Angle calculation is crucial when solving triangles, especially when given limited information like two sides and an angle. Here, using the Law of Sines, once we find one angle, it helps in finding others.

• The known angle \(\alpha = 42^{\circ}\) sets a starting point.
• To find angle \(\beta\), rearrange the formula to \(\frac{b}{\sin \beta} = \frac{c}{\sin \alpha}\). After substituting and solving for \(\sin \beta\), use a calculator to find \(\beta\) itself.
• Finally, calculate the third angle \(\gamma\) using the angle sum property of a triangle: \( \gamma = 180^{\circ} - \alpha - \beta \).

It's key to remember: the sum of a triangle's interior angles is always \(180^{\circ}\). This property guides the calculation when two angles are known, always leading to the third angle.

Inverse Sine Function

The inverse sine function, often denoted as \( \sin^{-1} \) or arcsin, is essential in finding angle measures from known sine values. When solving triangles, once we have a value for \( \sin \beta \), we need the inverse sine to extract the angle itself.

• Given \( \sin \beta \approx 0.889 \), calculate \( \beta \) as \( \beta = \sin^{-1}(0.889) \).
• Use a calculator: ensure it's in degree mode if working with degrees for accuracy.

This function is pivotal, turning a numeric sine value back into an angular measure, which is then used to finalize triangle solutions.