Summarize the following:
Suppose, as a different problem, that you know two sides and need to solve an unknown angle. You are given that side A is 10 inches long, side B is 7 inches long, and angle {\displaystyle lpha } is 50 degrees. You can use this information to find the measurement of angle {\displaystyle eta }. Set up the problem as follows: Asin=Bsin{\displaystyle {
rac {A}{\sin lpha }}={
rac {B}{\sin eta }}} 10sin50=7sin{\displaystyle {
rac {10}{\sin 50}}={
rac {7}{\sin eta }}} sin=7sin5010{\displaystyle \sin eta ={
rac {7\sin 50}{10}}} sin=70.76610{\displaystyle \sin eta ={
rac {7*0.766}{10}}} sin=0.536{\displaystyle \sin eta =0.536} In the above example, the law of sines provides the sine of the selected angle as its solution. To find the measure of the angle itself, you must use the inverse sine function. This is also called the arcsine. On a calculator, this is generally marked as sin1{\displaystyle \sin ^{-1}}. Use this to find the measure of the angle. For the example above, the final step is as follows: sin=0.536{\displaystyle \sin eta =0.536} =arcsin0.536{\displaystyle eta =rcsin 0.536} =32.4{\displaystyle eta =32.4}. Suppose you are told that angle =30degrees{\displaystyle lpha =30{ ext{ degrees}}}, angle =50degrees{\displaystyle eta =50{ ext{ degrees}}}, and side C, which connects them, is 10 inches long. Find the measurement of all sides and angles for the triangle. First, you should recognize that you do not yet have enough information for the sine rule to apply. The sine rule requires that you have at least one pair with an angle that opposes a known side. However, you can calculate the third angle of this triangle using simple subtraction. All three angles add up to 180 degrees, so you can find angle {\displaystyle \gamma } by subtracting: =180=1803050=100{\displaystyle \gamma =180-lpha -eta =180-30-50=100} Now that you know all three angles, you can use the sine rule to find the two remaining sides. Solve them one at a time: Csin=Bsin{\displaystyle {
rac {C}{\sin \gamma }}={
rac {B}{\sin eta }}} 10sin100=Bsin50{\displaystyle {
rac {10}{\sin 100}}={
rac {B}{\sin 50}}} 10sin50sin100=B{\displaystyle {
rac {10\sin 50}{\sin 100}}=B} 100.7660.985=B{\displaystyle {
rac {10*0.766}{0.985}}=B} 7.78=B{\displaystyle 7.78=B} Thus, side B is 7.78 inches long. Now solve for the final remaining side. Csin=Asin{\displaystyle {
rac {C}{\sin \gamma }}={
rac {A}{\sin lpha }}} 10sin100=Asin30{\displaystyle {
rac {10}{\sin 100}}={
rac {A}{\sin 30}}} 10sin30sin100=A{\displaystyle {
rac {10\sin 30}{\sin 100}}=A} 100.50.985=A{\displaystyle {
rac {10*0.5}{0.985}}=A} 5.08=A{\displaystyle 5.08=A} Side A, therefore, is 5.08 inches long. You now have all three angles, 30, 50 and 100 degrees, and all three sides, 5.08, 7.78, and 10 inches.
Solve for an unknown angle. Use the inverse function if needed to find the angle. Solve a problem with incomplete information.