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Chapter 5: Problem 20

Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$B=2^{\circ} 45^{\prime}, \quad b=6.2, \quad c=5.8$$

Short Answer

Expert verified
There are two possible solutions for the triangle: A = 125.83 degrees, B = 2.75 degrees, C = 51.42 degrees, a = 10.44, b = 6.2, c = 5.8 or A = 48.67 degrees, B = 2.75 degrees, C = 128.58 degrees, a = 4.26, b = 6.2, c = 5.8. Each value is rounded to two decimal places.

Step by step solution

01

Conversion of Angle to Decimal Form

First, convert the angle B from degrees and minutes to a decimal. The given angle is \(B= 2^{\circ}45^{\prime}\). There are 60 minutes in a degree, so to convert minutes to degrees, divide the minutes by 60. Therefore, \(B = 2 + \frac{45}{60} = 2.75^{\circ}\).
02

Applying the Law of Sines to Find Angle C

Now, apply the law of sines to find the unknown angle C. Using the formula \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\), solving for \(\sin(C)\) gives us \(\sin(C) = \frac{b}{c} \cdot \sin(B)\), in which b = 6.2, c = 5.8, and B = 2.75. Calculating this gives \(\sin(C) \approx 0.7861\). This gives two possible values for angle C, C = \(\arcsin(0.7861)\) and \(180 - \arcsin(0.7861)\). Taking the arc sine gives C = 51.42 or 128.58 degrees.
03

Finding Angle A

The sum of angles in a triangle is 180 degrees. You can find angle A by subtracting the other two angles from 180. Hence, \(A = 180 - B - C\). If C was chosen as 51.42, then \(A = 180 - 2.75 - 51.42 = 125.83^{\circ}\). If C was chosen as 128.58, then \(A = 180 - 2.75 - 128.58 = 48.67^{\circ}\)
04

Finding Side Length a

You can find the remaining side length a by using the law of sines. Using the formula \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\), solving for a gives us \(a = \frac{b}{\sin(B)} \cdot \sin(A)\). If angle A was chosen as 125.83, then \(a \approx 10.44\). If A was chosen as 48.67, then \(a \approx 4.26\).

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