Problem 11 - Precalculus with Limits

Question

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=83^{\circ} 20^{\prime}, \quad C=54.6^{\circ}, \quad c=18.1$$

Step-by-Step Solution

Answer

The missing angle B is about \(42.07^{\circ}\). The missing sides a and b are approximately 23.32 and 14.64 respectively.

Step 1: Calculate the Missing Angle

To solve a triangle means to know all three sides and all three angles. Given that the angles in a triangle always add up to 180 degrees, we can easily find the third angle B using the formula: \( B = 180 - A - C \). In this case, A is \(83^{\circ} 20^{\prime}\) which equal to \(83.33^{\circ}\) (since \(20' = 20/60 = 0.33\) degree), and C is \(54.6^{\circ}\). Therefore, \( B = 180 - 83.33 - 54.6 = 42.07^{\circ} \).

Step 2: Calculate the Missing Sides using Law of Sines

The Law of Sines indicates that the ratios for each side of a triangle to the sine of its respective angle are equal. That is, \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\). Given side that we know is c and its corresponding angle C, we can calculate sides a and b. Let's start with a. Based on the Law of Sines, we get \( a = c \cdot \frac{\sin(A)}{\sin(C)}\). Substituting the given values in, we have \(a = 18.1 \cdot \frac{\sin(83.33)}{\sin(54.6)}\). This calculation gives us approximately a = 23.32 to two decimal places. Similarly, we can calculate b using the same law. \( b = c \cdot \frac{\sin(B)}{\sin(C)}\), and substituting in the known values gives \(b = 18.1 \cdot \frac{\sin(42.07)}{\sin(54.6)}\), which gives us approximately b = 14.64 to two decimal places.

Key Concepts

Triangle Solving

When we solve a triangle, we aim to determine all its unknowns. This includes finding all sides and angles. For any triangle, the sum of its interior angles always equals 180 degrees. Thus, if you have two angles, finding the third is straightforward. This is a fundamental rule that aids in starting the process of solving triangles.
For example, with known angles \(A = 83^\circ 20'\) and \(C = 54.6^\circ\), we can find angle \(B\) by subtracting the sum of angles \(A\) and \(C\) from 180 degrees. Such calculations help us move forward to solve more parts of the triangle.

• Always remember the magical number: 180 degrees for triangle angles.
• Use subtraction to find the missing angle once two are known.
• Knowing all three angles gives a clear path to use other solving methods.

Angle Calculations

Angle calculations often require transforming minute values into decimals. In our example, \(20'\) is converted to a decimal, specifically \(0.33^\circ\), by dividing by 60 since a degree is made of 60 minutes. Such conversion simplifies calculations further.
When you're dealing with angles in trigonometric equations, being precise in measurements is crucial to obtaining accurate results.
This discipline also entails applying the relationships between the sides and angles, notably using trigonometric functions like sine, to solve for unknowns which we'll explore in greater detail in side calculations.

• Convert angle minutes to decimals for easy computation.
• Check if the angle measurements are entered correctly.
• Be precise with trigonometric functions and angle values.

Side Calculations

The Law of Sines is a pivotal method when it comes to side calculations in triangle solving. It relates the sides of a triangle to their opposite angles, making it a powerful tool. According to the Law of Sines, the ratio of a side's length to the sine of its opposite angle is constant for all three sides of a triangle. Therefore, \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).
This formula is the backbone of calculations in our exercise. We begin with known values, such as side \(c = 18.1\) and angle \(C = 54.6^\circ\), to find missing sides. For side \(a\), this involves computing \(a = 18.1 \cdot \frac{\sin(83.33)}{\sin(54.6)}\), yielding approximately 23.32.
Similarly, to find \(b\), we use \(b = 18.1 \cdot \frac{\sin(42.07)}{\sin(54.6)}\), arriving at approximately 14.64.

• The Law of Sines works best with two angles and one side known.
• Always match the angle to its opposite side when using the Law of Sines.
• Pay careful attention to your arithmetic and trigonometric steps.