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Chapter 5: Problem 26

Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=110^{\circ}, \quad a=125, \quad b=200$$

Short Answer

Expert verified
After following these steps, we will have the angles B, C and the length of side c for one or two possible solutions to the triangle. These are (or is) the solution(s) to the problem.

Step by step solution

01

Apply the Law of Sines to Calculate Angle B

The formula for the Law of Sines is: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \]. Substituting the given values: \[ \frac{125}{\sin(110^\circ)} = \frac{200}{\sin(B)} \]. Now, isolate \( \sin(B) \) on one side.
02

Find the first possible solution for Angle B

Solve for \( \sin(B) \) and find the arcsin to derive the value of the angle. Remember to round your answers to two decimal places.
03

Check if a second solution Exists

In a triangle, the sum of angles is \( 180^\circ \). If the value obtained in step 2 subtracted from \( 180^\circ \) and then added to \( 110^\circ \) (value of A) yields a sum less than \( 180^\circ \), then a second solution exists. If this is the case, calculate the second possible value for angle B by subtracting the first from \( 180^\circ \).
04

Calculate angle C and sides c for both solutions

The third angle can be found by subtracting the sum of the other two angles from \( 180^\circ \). Then use the law of sines again to find side c by rearranging the formula: \( c = \sin(C) * \frac{a}{\sin(A)} \). Remember to compute this for both solutions if a second solution exists.

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