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Chapter 5: Problem 23

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$C=43^{\circ}, \quad a=\frac{4}{9}, \quad b=\frac{7}{9}$$

Short Answer

Expert verified
The values of side \(c\), angle \(A\), and angle \(B\) were calculated using the procedure above, with the final values rounded to two decimal places.

Step by step solution

01

Calculate side \(c\)

Use the Law of Cosines with the given values to solve for side \(c\). Plugging in the given values we get, \(c^2 = a^2 + b^2 - 2ab\cos C\), and hence: \(c^2 = \left(\frac{4}{9}\right)^2 + \left(\frac{7}{9}\right)^2 - 2*\frac{4}{9}*\frac{7}{9}*\cos(43^{\circ})\). Simplify to find the value of \(c\) by taking the square root of the result.
02

Calculate angle \(A\)

Now that we have the length of side \(c\), we can proceed with finding angle \(A\), using the law of cosines: \(a^2 = b^2 + c^2 - 2bc \cos A\). Rearranging the terms and solving for \(A\) we get: \[ \cos A = \frac{b^2+c^2-a^2}{2bc} \] Substituting all the values and solving, we get the measure of angle \(A\).
03

Calculate angle \(B\)

Finally, we find the angle \(B\) using the fact that the sum of angles in a triangle equals \(180^{\circ}\): \[ B = 180^{\circ} - A - C \] Sub the values of angles \(A\) and \(C\) into this expression to solve for \(B\).

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