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Chapter 6: Problem 19

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=4, b=6, c=9$$

Short Answer

Expert verified
The triangle's angles, when calculated correctly and rounded to the nearest degree, are \[ A \approx 28^\circ \], \[ B \approx 51^\circ \] and \[ C \approx 101^\circ \].

Step by step solution

01

Calculation of the first angle

Let's use the Law of Cosines to find the angle A, opposite to side a. The formula is \( A = \arccos{\frac{b^2 + c^2 - a^2}{2bc}} \). Plugging in the given values yields: \( A = \arccos{\frac{6^2 + 9^2 - 4^2}{2*6*9}} \) which should be rounded to the nearest degree.
02

Calculation of the second angle

Next, let's use the Law of Sines to calculate the angle B, opposite to side b. The formula is \( B = \arcsin{\frac{b * \sin{A}}{a}} \). Substituting the given values and the previously found A result in, we have: \( B = \arcsin{\frac{6 * \sin{A}}{4}} \). Remember to round your result to the nearest degree.
03

Calculation of the third angle

Lastly, let's find the angle C using the fact that the sum of interior angles in a triangle is equal to 180 degrees. So, \( C = 180 - A - B \). Substitute A and B with their calculated values to get the value of C. Again, round the result to the nearest degree.

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