Problem 24 - Trigonometry

Question

Find all solutions to each of the following triangles: \(C=83.4^{\circ}, c=51.1 \mathrm{~km}, b=94.2 \mathrm{~km}\)

Step-by-Step Solution

Answer

The triangle has likely only one solution with approximate angles \( A \approx 68.01^{\circ} \), \( B \approx 28.59^{\circ} \), and \( C = 83.4^{\circ} \), and side \( a \).

Step 1: Review Given Information

We have a triangle with side \( c = 51.1 \, \text{km} \), side \( b = 94.2 \, \text{km} \), and an angle \( C = 83.4^{\circ} \). We need to find the remaining angles \( A \) and \( B \), and side \( a \).

Step 2: Use the Law of Sines

Using the Law of Sines: \( \frac{b}{\sin B} = \frac{c}{\sin C} \). Substituting the given values: \[ \frac{94.2}{\sin B} = \frac{51.1}{\sin 83.4^{\circ}} \]. Solve for \( \sin B \).

Step 3: Calculate \( \sin B \)

First, calculate \( \sin 83.4^{\circ} \). Then find \( \sin B \) using: \[ \sin B = \frac{94.2 \times \sin 83.4^{\circ}}{51.1} \].

Step 4: Find Angle \( B \)

Use the inverse sine function to find angle \( B \): \( B = \sin^{-1}(\sin B) \). Compute the value of \( B \).

Step 5: Determine Possible Triangle Cases

Since \( B \) and \( C \) sum to more than \( 90^{\circ} \), consider the possibility of two triangles (Ambiguous case) but verify it with given values. Confirm if only one triangle is possible.

Step 6: Calculate Angle \( A \)

Use the fact that the angles in a triangle add up to \( 180^{\circ} \): \[ A = 180^{\circ} - B - C \]. Solve to find \( A \).

Step 7: Use the Law of Sines to Find Side \( a \)

Now that we know \( A \), use the Law of Sines: \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Solve for \( a \).

Step 8: Confirm Solution with Cross-Verification

Verify the calculated values satisfy the initial triangle setup using Law of Sines for consistency across sides and angles.

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry, used to solve for unknown parts of a triangle when certain other parts are known. It is particularly useful when working with non-right triangles. The formula is straightforward:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Here, \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle, while \( A \), \( B \), and \( C \) are the angles opposite those sides, respectively.
To use this law effectively, you need to know:

• At least one side length and one angle, or
• Two angles and one side, or
• Two sides and a non-included angle.

In the exercise, we applied the Law of Sines to solve for an unknown angle \( B \), given \( b \) and \( c \), and angle \( C \). This involves substituting known values into the formula and solving for the unknown using basic algebraic manipulation and sine inverse functions.

Angle Calculation

Calculating angles in triangles often requires an understanding of the triangle's total angle sum. The angles in every triangle always sum up to \( 180^{\circ} \). Knowing this, you can find a missing angle when the other two are known through simple arithmetic.
In this exercise, after finding angle \( B \) using the Law of Sines, we determine angle \( A \) by subtracting the known angles \( B \) and \( C \) from \( 180^{\circ} \).
Furthermore, when solving for angles involving the sine function, you might encounter the ambiguous case scenario, particularly in triangles with obtuse or acute angles. This is when an angle from a sine value could correspond to two possible angles, because \( \sin(\theta) = \sin(180^{\circ} - \theta) \). It’s critical to verify the feasibility of both angles by considering the sum with the known angle, \( C \), to determine if the assumed total angles would form a valid triangle.

Triangle Solutions

Finding solutions to triangles involves determining all unknown angles and side lengths. To solve a triangle, you must apply both the knowledge of trigonometric laws and verify the constraints of triangle geometry.
The process for solving any triangle usually involves:

• Identifying what is known and what needs to be found.
• Using the Law of Sines or Law of Cosines to find unknown sides or angles.
• Ensuring the sum of angles equals \( 180^{\circ} \).
• Verifying that the side lengths adhere to triangle inequality principles, which state that the sum of any two sides must be greater than the third side.

In the provided exercise, you solve for two unknown angles and one unknown side. After determining all the angles, verify them against the sides using the Law of Sines, ensuring the triangle remains consistent and accurately described. Cross-verifying with calculations helps ensure that the triangle fulfills the given constraints and laws.