Question

solving oblique triangle problems using the law of sines
solve these problems by using the law of sines. compute side lengths to 3 decimal places, customary angular values to the nearest minute, and metric angular values to the nearest hundredth of a degree.

1. determine side a.
2. in a scalene triangle the measurement of two angles are 70.00 degrees and 33.00 degrees. the measurement of the side opposite angle 70 degrees is 9.800 centimeters and opposite angle 30 degrees is a.

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that for a triangle with sides \(a\), \(b\), \(c\) opposite angles \(A\), \(B\), \(C\) respectively, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\).
Here, we have angle \(A = 30.00^\circ\) (wait, correction: the problem says "opposite angle 30 degrees is a", and angle with opposite side 9.800 cm is \(70.00^\circ\). Wait, re - reading: "the measurement of two angles are 70.00 degrees and 33.00 degrees. The measurement of the side opposite angle 70 degrees is 9.800 centimeters and opposite angle 30 degrees is a." Wait, maybe a typo, maybe the angle opposite \(a\) is \(33.00^\circ\)? Wait, the diagram has 70.00 and 33.00 degrees. Let's assume: let angle \(A = 33.00^\circ\), angle \(B=70.00^\circ\), side \(b = 9.800\) cm (opposite angle \(B\)), and side \(a\) opposite angle \(A\).
So by Law of Sines, \(\frac{a}{\sin A}=\frac{b}{\sin B}\)

Step2: Solve for \(a\)

We can re - arrange the formula to \(a=\frac{b\times\sin A}{\sin B}\)
Given \(b = 9.800\) cm, \(A = 33.00^\circ\), \(B = 70.00^\circ\)
First, calculate \(\sin(33.00^\circ)\approx0.5446\), \(\sin(70.00^\circ)\approx0.9397\)
Then \(a=\frac{9.800\times\sin(33.00^\circ)}{\sin(70.00^\circ)}=\frac{9.800\times0.5446}{0.9397}\)
Calculate numerator: \(9.800\times0.5446 = 9.8\times0.5446=5.33708\)
Then divide by \(0.9397\): \(a=\frac{5.33708}{0.9397}\approx5.680\) (wait, maybe the angle opposite \(a\) is \(30^\circ\)? Wait, the problem says "opposite angle 30 degrees is a". Let's re - check. If angle opposite \(a\) is \(30^\circ\), angle opposite 9.800 is \(70^\circ\)
Then \(a=\frac{9.800\times\sin(30^\circ)}{\sin(70^\circ)}\)
\(\sin(30^\circ)=0.5\), \(\sin(70^\circ)\approx0.9397\)
\(a=\frac{9.800\times0.5}{0.9397}=\frac{4.9}{0.9397}\approx5.214\). But the diagram has 70 and 33. Maybe the angle opposite \(a\) is \(33^\circ\), angle opposite 9.8 is \(70^\circ\)
So \(a=\frac{9.8\times\sin(33^\circ)}{\sin(70^\circ)}\)
\(\sin(33^\circ)\approx0.5446\), \(\sin(70^\circ)\approx0.9397\)
\(a=\frac{9.8\times0.5446}{0.9397}=\frac{5.33708}{0.9397}\approx5.680\) (wait, maybe the two angles are 70 and 33, and the third angle is \(180-(70 + 33)=77\), but the problem says "opposite angle 30 degrees is a" which might be a typo. Assuming that the angle opposite \(a\) is \(33^\circ\), angle opposite 9.8 is \(70^\circ\))

Answer:

\(a\approx5.680\) cm (if we take angle opposite \(a\) as \(33^\circ\)) or if angle opposite \(a\) is \(30^\circ\), \(a\approx5.214\) cm. But based on the diagram with 70 and 33, the correct calculation with angle opposite \(a = 33^\circ\), \(a=\frac{9.8\times\sin(33^\circ)}{\sin(70^\circ)}\approx5.68\) cm (rounded to 3 decimal places).