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Chapter 5: Problem 17

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$B=125^{\circ} 40^{\prime}, \quad a=37, \quad c=37$$

Short Answer

Expert verified
The completed triangle has measures \(b \approx 60.54\), \(A \approx 34.45^{\circ}\), and \(C \approx 19.88^{\circ}\).

Step by step solution

01

Convert Given Angle to Decimal

First, convert \(B = 125^{\circ} 40^{\prime}\) to decimal. Remember that there are 60 minutes in a degree, so to convert, divide the minutes by 60 and add the result to the degrees. So, \(B = 125.67^{\circ}\).
02

Use Law of Cosines to Find Side Length

Then, use the Law of Cosines to find the length of side b. The Law of Cosines states that, for any triangle with sides of lengths a, b, and c and with the angle B positioned as indicated, the equation \(b^2 = a^2 + c^2 - 2ac\cos(B)\) holds true. Therefore, \(b = \sqrt{(37^2 + 37^2 - 2 * 37 * 37 \cos(125.67))}\), which gives \(b \approx 60.54\).
03

Use Law of Sines to Find Angle A

Next, use the Law of Sines to find angle A. This law states that the ratio of a side length to the sine of its opposite angle is constant in a triangle. So, \(\sin(A) = \frac{a}{b} \sin(B)\). Solving gives \(A = \sin^{-1}(\frac{37}{60.54} \sin(125.67))\), and so \(A \approx 34.45^{\circ}\).
04

Find the Last Angle

Find the last angle C by knowing that the sum of all angles in a triangle add up to 180. So, \(C = 180 - B - A\), which gives \(C \approx 19.88^{\circ}\).

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