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Chapter 6: Problem 12

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$A=85^{\circ}, B=35^{\circ}, c=30$$

Short Answer

Expert verified
The solution to the exercise is \( C = 60^{\circ} \), Side \( a \approx 32.9 \) units, and Side \( b \approx 17.3 \) units.

Step by step solution

01

Finding Angle C

Using the Triangle Angle Sum Theorem, we know that \( A + B + C = 180 \) degrees. Given \( A = 85^{\circ} \) and \( B = 35^{\circ} \), we can calculate angle \( C \) as \( C = 180 - A - B = 180 - 85 - 35 = 60^{\circ} \).
02

Use The Law of Sines to Find Side a

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. So, \( \frac{a}{\sin A} = \frac{c}{\sin C} \). We will isolate \( a \) to find its value: \( a = \frac{c \sin A}{\sin C} = \frac{30 \sin 85^{\circ} }{\sin 60^{\circ} } \).
03

Use The Law of Sines to Find Side b

Similarly, we can find side b by setting up the equation as \( \frac{b}{\sin B} = \frac{c}{\sin C} \) and isolating \( b \), it will be: \( b = \frac{c \sin B}{\sin C} = \frac{30 \sin 35^{\circ} }{\sin 60^{\circ} } \).

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