Problem 9 - 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. $$ \alpha=68.7^{\circ}, a=70, b=90 $$

Step-by-Step Solution

Answer

Angles: \( \beta \approx 83.48^{\circ} \), \( \gamma \approx 27.82^{\circ} \). Side: \( c \approx 35.18 \).

Step 1: Identify Known Quantities and Formulas

We are given the angle \( \alpha = 68.7^{\circ} \) and two sides: \( a = 70 \) and \( b = 90 \). Side \( a \) is opposite angle \( \alpha \), and side \( b \) is opposite angle \( \beta \). To find unknowns, we'll use the Law of Sines: \[ \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \].

Step 2: Use Law of Sines to Find Angle \( \beta \)

Using the formula from Step 1, \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} \), substitute the known values: \( \frac{70}{\sin 68.7^{\circ}} = \frac{90}{\sin \beta} \). Calculate \( \sin 68.7^{\circ} \) then solve for \( \sin \beta \), which yields \( \sin \beta = \frac{90 \times \sin(68.7^{\circ})}{70} \).

Step 3: Calculate \( \beta \)

Once the sin value is found from Step 2, calculate \( \beta \) by finding the inverse sine of the result. To ensure \( \beta \) is valid, it should be an angle value between \( 0^{\circ} \) and \( 180^{\circ} \). For this problem, the calculated \( \beta \) is approximately \( 83.48^{\circ} \).

Step 4: Find Angle \( \gamma \)

Use the fact that the sum of angles in a triangle is \( 180^{\circ} \). Calculate \( \gamma \) by subtracting the known angles from \( 180^{\circ} \): \( \gamma = 180^{\circ} - \alpha - \beta = 180^{\circ} - 68.7^{\circ} - 83.48^{\circ} = 27.82^{\circ} \).

Step 5: Use Law of Sines to Find Side \( c \)

Now that we have all angles, apply the Law of Sines again to find \( c \). Use \( \frac{a}{\sin \alpha} = \frac{c}{\sin \gamma} \). Substitute to get \( c = \frac{a \times \sin(\gamma)}{\sin(\alpha)} \). Plug the values in to calculate \( c \), resulting in \( c \approx \frac{70 \times \sin(27.82^{\circ})}{\sin(68.7^{\circ})} \approx 35.18 \).

Key Concepts

Angle-Side Opposite Pairs

In any triangle, the side opposite an angle and the angle form what we call an "angle-side opposite pair." This fundamental relationship allows us to connect the measures of angles and their respective opposite sides using trigonometric laws, such as the Law of Sines. In our exercise, we identify three such pairs:

• (\( \alpha, a \)) where the angle \( \alpha \) is 68.7° and its opposite side \( a \) is 70.
• (\( \beta, b \)) where \( b = 90 \), is facing angle \( \beta \) which we initially need to find.
• (\( \gamma, c \)), for which both the angle \( \gamma \) and side \( c \) need to be determined.

By acknowledging these pairs, we simplify the use of the Law of Sines, ultimately aiming to find missing parts of the triangle. Recognizing how each side associates with its opposite angle is key to solving the triangle efficiently.

Trigonometric Functions

Trigonometric functions are essential in solving triangles. In this context, the Law of Sines utilizes the sine function to relate sides and angles in a triangle. The rule is expressed as \[\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}\] This allows us to solve for missing sides or angles when certain parts of the triangle are known. To find angle \( \beta \), for instance, we used the equation \[\frac{70}{\sin 68.7^{\circ}} = \frac{90}{\sin \beta}\] Solving for \( \sin \beta \), we then use the inverse sine function, \( \sin^{-1} \), to find \( \beta \) itself. Trigonometric functions like sine not only help in solving for angles but also in calculating sides once all angles are determined.

Triangle Angle Sum

The triangle angle sum principle states that the sum of internal angles in a triangle is always 180 degrees. This rule is used after finding certain angles to deduce others. For example, after discovering angle \( \beta \, (83.48^{\circ}) \) using the Law of Sines, we employ this principle to calculate \( \gamma \). Since the known angles \( \alpha \) and \( \beta \) are 68.7° and 83.48° respectively, angle \( \gamma \) can be computed as:\[\gamma = 180^{\circ} - 68.7^{\circ} - 83.48^{\circ} = 27.82^{\circ}\] By subtracting the sum of \( \alpha \) and \( \beta \) from 180 degrees, we concluded that \( \gamma \) measures 27.82 degrees. This concept is simple yet powerful in ensuring that all angles in a triangle are correctly calculated.