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Chapter 5: Problem 11

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

Short Answer

Expert verified
The final measurements of the triangle are: Angle B, in degrees; side a, in units; and side b, in units.

Step by step solution

01

Find Angle B

Using the property that the sum of interior angles in a triangle is 180 degrees, calculate angle B using the formula: \[ B = 180^{\circ} - (A + C) \] Hence, \( B = 180^{\circ} - (83^{\circ} 20^{\prime} + 54.6^{\circ}) \)
02

Convert Angle B to Decimal

In order to work more efficiently with the measurements, convert the minutes in angle A to decimal form. Then convert angle B from degrees to decimal. Each degree consists of 60 minutes, so divide 20 by 60 and add this to 83. Then use this value to calculate angle B in decimal.
03

Apply the Law of Sines

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. Using this law, find sides a and b using the formulas: \[ a = c \times \frac{\sin(A)}{\sin(C)} \] and \[ b = c \times \frac{\sin(B)}{\sin(C)} \]
04

Compute the Values

Substitute the given and derived values (from Steps 1, 2 and 3) into the respective formulas to compute for the lengths of sides a and b.

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