Trigonometry : Ambiguous Triangles

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : Ambiguous Triangles

Solve for . Image not drawn to scale. There may be more than one answer.

Ambiguous triangle 1

Possible Answers:

Correct answer:

Explanation:

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.

Example Question #1 : Ambiguous Triangles

Solve for . Image not drawn to scale; there may be more than one solution.

Ambiguous triangle 2

Possible Answers:

Correct answer:

Explanation:

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 4.

 Solve for x by evaluating  in a calculator.

There is another solution as well. If  has a sine of 0.951, so will its supplementary angle,

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

Example Question #3 : Ambiguous Triangles

If  = , and  find  to the nearest degree.

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #4 : Ambiguous Triangles

If , find  to the nearest tenth of a degree.

Possible Answers:

and 

and 

Correct answer:

and 

Explanation:

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 .

 

 

Example Question #5 : Ambiguous Triangles

If , and  =  find  to the nearest degree.

Possible Answers:

 and 

 and 

Correct answer:

 and 

Explanation:

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 angle . 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 of the problem

Rearranging the equation to isolate 

 

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 

Example Question #6 : Ambiguous Triangles

If c=10.3a=7.4, and  find  to the nearest degree.

Possible Answers:

 and 

No solution

Correct answer:

No solution

Explanation:

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 angle . In this case, there may be one or two triangles determined. If , then the  is not a solution.

In this problem, , which means that there are no solutions to  that satisfy this triangle. If you got answers for this triangle, check that you set up your Law of Sines equation properly at the start of the problem. 

Example Question #7 : Ambiguous Triangles

If , and  =  find  to the nearest degree.

Possible Answers:

 and 

No solution

Correct answer:

Explanation:

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 angle . 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 of the problem

Rearranging the equation to isolate 

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 greater than the other given side . Therefore, we have only one solution, .

Example Question #8 : Ambiguous Triangles

If , and  find  to the nearest degree.

Possible Answers:

No solution

 and

 and 

Correct answer:

Explanation:

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 angle . 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 of the problem

Rearranging the equation to isolate 

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

  • No solution when the side opposite the given angle is less than or equal to the other given side
  • One solution if the side opposite the given angle is greater than the other given side

In this problem, the side opposite the given angle is , which is greater than the other given side . Therefore this problem has one and only one solution, 

Example Question #9 : Ambiguous Triangles

If , and  =  find  to the nearest degree.

Possible Answers:

No solution

 and 

 and 

Correct answer:

No solution

Explanation:

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 angle . In this case, there may be one or two triangles determined. If , then the  is not a solution.

In this problem, , which means that there are no solutions to  that satisfy this triangle. If you got answers for this triangle, check that you set up your Law of Sines equation properly at the start of the problem. 

Example Question #10 : Ambiguous Triangles

If c=70a=50, and  find  to the nearest degree.

Possible Answers:

 and 

no solution

 and 

Correct answer:

no solution

Explanation:

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 angle . In this case, there may be one or two triangles determined. If , then the  is not a solution.

In this problem, , which means that there are no solutions to  that satisfy this triangle. If you got answers for this triangle, check that you set up your Law of Sines equation properly at the start of the problem. 

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