The Law of Sines is a set of ratios that allows one to compute missing angles and side lengths of oblique triangles (non right angle triangles).

The Law of Sines: 

.

To find the missing angle A we subtract the sum of the two known angles from 180° as the interior angles of all triangles equal 180°.

The LOS can be rearranged to solve for the missing side.

This is a straightforward Law of Sines problem as we are given two angles and a corresponding side:

Substituting the known values:

Solving for the unknown side:

This is a straightforward Law of Sines problem since we are given one angle and two sides and are asked to determine the corresponding angle.

Substituting the given values:

Now rearranging the equation:

The final step is to take the inverse sine of both sides:

We are given two angles and the length of the corresponding side to one of those angles. Because the problem is asking for the corresponding length of the other angle we can use the Law of Sines to find the length of the side . The equation for the Law of Sines is 

If we rearrange the equation to isolate  we obtain

Substituting on the values given in the problem

Because we are given the two angles and the length of the corresponding side to one of those angles, we can use the Law of Sines to find the length of the side that we need. So we use the equation

Rearranging the equation to isolate  gives

Substituting in the values from the problem gives

We can use the Law of Sines to find the length of the missing side, because we have its corresponding angle and the length and angle of another side. The equation for the Law of Sines is 

Isolating  gives us

Finally, substituting in the values of the of from the problem gives

To solve, use Law of Sines, , where A is the angle across from side a, and B is the angle across from side b. In this case, our proportion is set up like this:

cross-multiply

evaluate the right side using a calculator

divide both sides by 7

solve for x by evaluating in a calculator

There is another solution as well. If has a sine of 0.734, so will its supplementary angle, . 

Since is still less than , is a possible value for x.

Notice that the given information is Angle-Side-Side, which is the ambiguous case. Therefore, we should test to see if there are no triangles that satisfy, one triangle that satisfies, or two triangles that satisfy this. From , we get . In this equation, if , no angle A that satisfies the triangle can be found. If ,  and there is a right triangle determined. Finally, if , two measures of angle B can be calculated:  an acute angle B and an obtuse angle . In this case, there may be one or two triangles determined. If , then the angle B' is not a solution.

In this problem, ,  and there is one right triangle determined. Since we have the length of two sides of the triangle and the corresponding angle of one of the sides, we can use the Law of Sines to find the angle that we are looking for. This goes as follows:

Inputting the lengths of the triangle into this equation

Isolating 

Notice that the given information is Angle-Side-Side, which is the ambiguous case. Therefore, we should test to see if there are no triangles that satisfy, one triangle that satisfies, or two triangles that satisfy this. From , we get . In this equation, if , no  that satisfies the triangle can be found. If ,  and there is a right triangle determined. Finally, if , two measures of  can be calculated: an acute  and an obtuse . In this case, there may be one or two triangles determined. If , then the  is not a solution.

In this problem, , so there may be one or two angles that satisfy this triangle. Since we have the length of two sides of the triangle and the corresponding angle of one of the sides, we can use the Law of Sines to find the angle that we are looking for. This goes as follows:

Inputting the values from the problem

When the original given angle () is acute, there will be:

• One solution if the side opposite the given angle is equal to or greater than the other given side
• No solution, one solution (right triangle), or two solutions if the side opposite the given angle is less than the other given side

In this problem, the side opposite the given angle is , which is less than the other given side . Therefore, we have a second solution. Find it by following the below steps:

, so  is a solution.

Therefore there are two values for an angle, and .