Question

Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. (If not possible, enter IMPOSSIBLE.) A = 72°, a = 34, b = 21

Answer 1

`B\approx35.97^\circ\\C\approx72.03^\circ\\c\approx34`

Explanation:

Law of Sines

`(sinA)/(a)=(sinB)/(b)=(sinC)/(c)`

Given information

`A=72^\circ\\a=34\\b=21`

Check if solutions exist

As
`A=72^\circ < 90^\circ` and that
`a > b\rightarrow 34 > 21`, then there exists only one possible triangle by the Ambiguous Case

Solve the triangle

`(sin(72^\circ))/(34)=(sin(B))/(21)\\ \\ 21sin(72^\circ)=34sin(B)\\\\(21sin(72^\circ))/(34)=sin(B)\\ \\B=sin^(-1)((21sin(72^\circ))/(34))\\ \\B=35.97394255^\circ\approx35.97^\circ`

`A+B+C=180^\circ\\\\72^\circ+35.97394255^\circ+C=180^\circ\\\\107.97394255^\circ+C=180^\circ\\\\C=72.02605745^\circ\approx72.03^\circ`

`(sin(72^\circ))/(34)=(sin(72.02605745^\circ))/(c)\\\\c*sin(72^\circ)=34sin(72.02605745^\circ)\\\\c=(34sin(72.02605745^\circ))/(sin(72^\circ))\\ \\c=34.00502065\approx34`

Answer 2

Given: A = 72°, a = 34, b = 21

Calculated: B = 35.97°, C = 72.03°, c = 34.00

Explanation:

`(\sin A)/(a) = (\sin B)/(b)`

`(\sin 72^\circ)/(34) = (\sin B)/(21)`

`\sin B = (21\sin 72^\circ)/(34)`

`\sin B = 0.5874`

`B = \sin^(-1) 0.5874`

`B = 35.97^\circ`

C = 180° - 72° - 35.97°

C = 72.03°

`(\sin A)/(a) = (\sin C)/(c)`

`(\sin 72^\circ)/(34) = (\sin 72.03^\circ)/(c)`

`c = (\sin 72.03^\circ * 34)/(\sin 72^\circ)`

`c = 34.00`

Given: A = 72°, a = 34, b = 21

Calculated: B = 35.97°, C = 72.03°, c = 34.00