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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 dimensions of the triangle are \(B = 42.07^{\circ}\), \(a = 22.74\), and \(b = 15.06\).

Step by step solution

01

Convert given data

Convert the given angle A from degrees and minutes to decimal form. For \(A=83^{\circ} 20^{\prime}\), convert minutes to degrees. Since there are 60 minutes in a degree, \(20 \,minutes = \frac{20}{60} = 0.33 \,degrees\). Therefore, \(A = 83.33^{\circ}\)
02

Calculate angle B

Find angle \(B\) from the given angles \(A\) and \(C\). The sum of angles in a triangle equals \(180\) degrees. Hence, \(B = 180 - A - C = 180 - 83.33 - 54.6 = 42.07^{\circ}\)
03

Calculate side a

Apply the law of sines to calculate the length of side \(a\) opposing angle \(A\). The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite is same for all three sides. Hence, \(\frac{a}{sinA} = \frac{c}{sinC}\). Therefore, \(a = c \cdot \frac{sinA}{sinC} = 18.1 \cdot \frac{sin83.33}{sin54.6} = 22.74\)
04

Calculate side b

Similarly, calculate the length of side \(b\) opposing angle \(B\). Hence, \(b = c \cdot \frac{sinB}{sinC} = 18.1 \cdot \frac{sin42.07}{sin54.6} = 15.06\)

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Most popular questions from this chapter

Consider the equation \(2 \sin x-1=0\). Explain the similarities and differences between finding all solutions in the interval \(\left[0, \frac{\pi}{2}\right)\), finding all solutions in the interval \([0,2 \pi),\) and finding the general solution.

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