Problem 16 - 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. $$B=28^{\circ}, \quad C=104^{\circ}, \quad a=3 \frac{5}{8}$$

Step-by-Step Solution

Answer

The dimensions of the triangle are A = 48 degrees, b = 2.37, c = 5.34.

Step 1: Convert Side a into Decimal

Instead of using 3 5/8, which is a mixed number, convert it to a decimal for easier computation. Adding the whole number 3 to the fraction 5/8 gives the decimal 3.625.

Step 2: Calculate Angle A

The sum of the angles in a triangle is equal to 180 degrees. Use this knowledge to calculate angle A by subtracting the sum of angle B and angle C from 180. This gives : \( A = 180 - 28 - 104 = 48 \, degrees \)

Step 3: Calculate Side b

Apply the law of sines to calculate for side length b. You should compare the ratio of side a and its opposite angle A to side b and its opposite angle B. Rearrange the equation to solve for b: \( b = (a \cdot sin(B)) / sin(A) \). Substituting the found and given values gives \( b = (3.625 \cdot sin(28)) / sin(48) \). Evaluate this to find that \( b = 2.37 \)

Step 4: Calculate Side c

Similarly, apply the law of sines to calculate for side length c. This time, compare the ratio of side a and its opposite angle A to side c and its opposite angle C. Rearrange the equation to solve for c: \( c = (a \cdot sin(C)) / sin(A) \). Substituting the found and given values gives \( c = (3.625 \cdot sin(104)) / sin(48) \). Evaluate this to find that \( c = 5.34 \)

Key Concepts

Triangle

A triangle is a polygon with three edges and three vertices. This simple shape is a fundamental building block in geometry. In any triangle, the sum of the internal angles is always 180 degrees. Triangles are classified by their angles and sides into several types, such as equilateral, isosceles, and scalene triangles. In our problem, we have an oblique triangle because it does not have a right angle. Understanding the properties of triangles, including how their angles and sides relate, is crucial for solving problems like the one presented using the Law of Sines.
The Law of Sines is particularly useful for oblique triangles. It connects the ratios of the lengths of sides to the sines of their opposite angles.

Angle Calculation

Angle calculation within a triangle involves finding unknown angles or checking for consistency among known ones. In our example, two angles were given: B and C. Since the sum of all angles in a triangle is always 180 degrees, we can find the missing angle A using the formula:

• Add the known angles: B + C = 28 + 104 = 132 degrees
• Subtract from 180: A = 180 - 132 = 48 degrees

This simple subtraction provides the missing angle, completing our triangle.

Decimal Conversion

Decimal conversion is important for handling calculations easily and accurately, especially when dealing with mixed numbers. In this exercise, side a was initially given as the mixed number 3 5/8. To convert this into a decimal:

• First, convert the fraction 5/8, which equates to 0.625.
• Then, add the whole number part, 3, to get the decimal form 3.625.

This conversion simplifies calculations with the Law of Sines, easing the process of using trigonometric functions and providing greater accuracy.

Side Calculation

Calculating the sides of a triangle can be efficiently done using the Law of Sines, when two angles and one side are known. This law helps set up an equation where one side of a triangle corresponds to the sine of its opposite angle. Here’s how it works:

• For finding side b: use the formula \( b = \frac{a \cdot \sin(B)}{\sin(A)} \)
• For finding side c: use the formula \( c = \frac{a \cdot \sin(C)}{\sin(A)} \)

You already know the values of angle A, B, and C, and side a. By substituting these into the formulas and calculating, you determine side b as 2.37 and side c as 5.34. Precision in these steps ensures the triangle is solved correctly.