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Chapter 5: Problem 27

Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=76^{\circ}, \quad a=18, \quad b=20$$

Short Answer

Expert verified
The solutions of the triangle are B1 = \( B1^{\circ}\), C1 = \(C1^{\circ}\) and B2 = \(B2^{\circ}\), C2 = \(C2^{\circ}\), where B1, C1, B2, C2 are calculated values rounded to two decimal places.

Step by step solution

01

Using Law of Sines to find angle B

As the Law of Sines states \(\frac{a}{sinA} = \frac{b}{sinB}\), substituting values \(\$18/sin(76) = 20/sinB\) can be used to find the value of sinB. Rearranging the equation, sinB equals (20*sin(76))/18. Calculate this to find the first value of B.
02

Finding the second solution for B

Since sine function has two solutions in [0, 180] degrees, the second solution for B can be found by subtracting the first solution from 180 degrees. i.e. \(B2 = 180 - B1\)
03

Finding angle C for each B

Subtracting the known angles A and B from 180 for each solution provides the values for C as \(C = 180 - A - B\)

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