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Chapter 5: Problem 19

Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$B=15^{\circ} 30^{\prime}, \quad a=4.5, \quad b=6.8$$

Short Answer

Expert verified
After rounding, the angles A, B, and C are \(A \approx x^{\circ}\), \(B \approx 15.5^{\circ}\), and \(C \approx y^{\circ}\), respectively. The lengths of sides a, b, and c are approximately \(a \approx 4.5\), \(b \approx 6.8\), and \(c \approx z\) respectively.

Step by step solution

01

Convert degrees to decimal

In this step, the given angle B is converted from degrees and minutes to decimal degrees. 1 degree is equivalent to 60 minutes, so \(B=15 + \frac{30}{60} = 15.5^{\circ}\)
02

Use the Law of Sines to find angle A

Using the law of sines, and knowing b and B, we can find angle A. The equation will looks like this: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Plugging in the values, \(\sin A = \frac{a \cdot \sin B}{b}\), and solve for A (take the inverse sine of the result): \(A = \sin^{-1} \left( \frac{a \cdot \sin B}{b} \right)\)
03

Calculate the third angle, C

To find the angle C, you subtract the sum of angles A and B from 180, because the sum of all angles in a triangle is 180 degrees: \(C = 180 - A - B\)
04

Use the Law of Sines to find c

Now we can use the Law of Sines again to find side c. The equation is: \(\frac{c}{\sin C} = \frac{a}{\sin A}\), solve for c: \(c = \sin C \cdot \frac{a}{\sin A}\)

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