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Chapter 6: Problem 10

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$A=56^{\circ}, C=24^{\circ}, a=22$$

Short Answer

Expert verified
The length of side c, and the measures of angles A and B should be computed using the steps. Let's note that actual values depend on the calculations.

Step by step solution

01

Identify Given Variables

The variables given are \( a=10 \), \( b=3 \), and \( C=15^{\circ} \). These represent two side lengths and the angle between them in a triangle.
02

Use Law Of Cosines To Find Side c

By applying the law of cosines, which states \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \), we can solve for side c. Substitute the given values in: \( c^{2}=10^{2}+3^{2}-2*10*3\cos(15) \) to obtain the length of side c.
03

Use Law Of Sines To Find Angles A and B

The law of sines states \( \frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)} \). Using the values found in the previous steps, the sides and angle C, we can compute the value of angles A and B respectively. For instance, to find angle A, rearrange the formula to: \( A=\arcsin(\frac{a}{c}\sin(C)) \), substitute the known values and calculate angle A. Repeat the process for angle B using \( B=\arcsin(\frac{b}{c}\sin(C)) \) ensuring that the angles add up to \( 180^{\circ} \).
04

Round Final Values

The problem asks for the lengths to be rounded to the nearest tenth and angle measures to the nearest degree. Make sure the final answers follow these rounding instructions.

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