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Chapter 5: Problem 18

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$A=48^{\circ}, \quad b=3, \quad c=14$$

Short Answer

Expert verified
After performing the above calculations, the values for the unknown side and angles in the triangle are found and rounded to two decimal places where applicable.

Step by step solution

01

Calculate Side Length a

Since we know angle \(A\) and sides \(b\) and \(c\), use the law of cosines to solve for \(a\). The equation will be: \(a = \sqrt{b^2 + c^2 -2bc \cos(A)}\). Substitute \(A=48^{\circ}, \quad b=3, \quad c=14\) into the equation, then calculate. Make sure to convert the degrees into radians before calculation.
02

Calculate Angle B

Once we have calculated side \(a\), we can then use the law of cosines to solve for angle \(B\). The formula in terms of \(B\) is \(b^2 = a^2 + c^2 -2ac \cos(B)\). Rearrange the equation to calculate \(B\), \(B = \cos^{-1}\left[(a^2 + c^2 - b^2) / (2ac)\right]\). Substitute the values of \(a\), \(b\), \(c\) into the equation and solve for \(B\). Remember that \(B\) will be in radians and must be converted to degrees.
03

Calculate Angle C

Since we are dealing with a triangle, the sum of the internal angles of a triangle is equal to \(180\) degrees. Now that we know both \(A\) and \(B\), we can easily calculate \(C\) using the equation \(C = 180 - A - B\). Substitute the calculated values of \(A\) and \(B\) into the equation to get \(C\).

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