The measures of two sides of a triangle and the angle opposite one of them are given. Determine the number of triangles that satisfy the given set of conditions. Please show your solutions.

Question

b = 27, a = 46, β = 38°

Answer 1

Let us examine the possible triangle arrangements with the information we know.  We have two side lengths (a=46, b=27) and one angle measurement (∠38°).  Begin by drawing several blank triangles.  Label the triangle vertices A, B and C.  The following side / angle relationships MUST be maintained in each diagram…

     •Side “a” is opposite ∠A

     •Side “b” is opposite ∠B

     •Side “c” is opposite ∠C

Given that the known angle must be opposite one of the two known sides, we have 4 possible arrangements to check.  Fill-in the following information on each triangle…

Case 1a:  m∠A=38°

Side “a=46” is opposite ∠A

Side “b=27” is opposite ∠B; m∠B is unknown

m∠C, side opposite m∠C are both unknown

Case 1b:  m∠A=38°

Side “a=27” is opposite ∠A

Side “b=46” is opposite ∠B; m∠B is unknown

m∠C, side opposite m∠C are both unknown

Case 2a:  m∠B=38°

Side “a=46” is opposite ∠A; m∠A is unknown

Side “b=27” is opposite ∠B

m∠C, side opposite ∠C are both unknown

Case 2b:  m∠B=38°

Side “a=27” is opposite ∠A; m∠A is unknown

Side “b=46” is opposite ∠B

m∠C, side opposite ∠C are both unknown

Let us examine each case illustrated above:

Case 1a:

Recognize we have a known side length opposite a known angle measurement and an adjacent known side length.  By definition, we can apply the Law of Sines which is [(SIN∠A)/a = (SIN∠B)/b = (SIN∠C)/c].  We substitute our values for this case…

     (SIN∠A)/a = (SIN∠B)/b

 (SIN 38°)/46 = (SIN∠B)/27

       0.62/46 = (SIN∠B)/27

    46(SIN∠B) = 0.62(27)

    46(SIN∠B) = 16.62

          m∠B = SIN⁻¹(16.62/46)

          m∠B = SIN⁻¹(0.36)

         ∴ m∠B = 21.18°

We can solve for the remaining side and angle…

       ∴ m∠C = 120.82°

           ∴ c = 64.17

…and can, therefore, conclude Case1a is a possible triangle arrangement.  Check the results by making sure all Law of Sines ratios are equal.

Case 1b:

Again, recognize that we have a known side length opposite a known angle measurement and an adjacent known side length.  By definition, we can again apply the Law of Sines.  We substitute our values for this case…

     (SIN∠A)/a = (SIN∠B)/b

 (SIN 38°)/27 = (SIN∠B)/46

       0.62/27 = (SIN∠B)/46

    27(SIN∠B) = 0.62(46)

    27(SIN∠B) = 28.32

          m∠B = SIN⁻¹(28.32/27)

          m∠B = SIN⁻¹(1.05)

        ∴ m∠B = ERROR, NO SOLUTION!!

We can conclude Case 1b is NOT a possible triangle arrangement.

Case 2a:

Again, recognize that we have a known side length opposite a known angle measurement and an adjacent known side length.  By definition, we can again apply the Law of Sines.  We substitute our values for this case…

    (SIN∠B)/b = (SIN∠A)/a

 (SIN 38°)/27 = (SIN∠A)/46

      0.62/27 = (SIN∠A)/46

    27(SIN∠A) = 0.62(46)

    27(SIN∠A) = 28.32

          m∠A = SIN⁻¹(28.32/27)

          m∠A = SIN⁻¹(1.05)

        ∴ m∠A = ERROR, NO SOLUTION!!

We can conclude Case 2a is NOT a possible triangle arrangement.  The calculation is identical to Case 1b above with angle and side values re-assigned.

Case 2b:

Again, recognize that we have a known side length opposite a known angle measurement and an adjacent known side length.  By definition, we can again apply the Law of Sines.  We substitute our values for this case…

    (SIN∠B)/b = (SIN∠A)/a

 (SIN 38°)/46 = (SIN∠A)/27

       0.62/46 = (SIN∠A)/27

    46(SIN∠A) = 0.62(27)

    46(SIN∠A) = 16.62

          m∠A = SIN⁻¹(16.62/46)

          m∠A = SIN⁻¹(0.36)

        ∴ m∠A = 21.18°

We can solve for the remaining side and angle…

       ∴ m∠C = 120.82°

           ∴ c = 64.17

…and can, therefore, conclude Case 2b is a possible triangle arrangement.  Check the results by making sure all Law of Sines ratios are equal.  Note: The calculation is identical to Case 1a above with angle and side values re-assigned.

In summary:

Of the four possible triangle arrangement cases presented, two cases (cases 1a and 2b) produce possible triangle arrangements.  Therefore, “2 possible triangle arrangements“ is the accurate answer to the problem statement. 

There are Law of Sines (and Cosines) calculators online which we can use to check our solutions.  Always check your work!

Answer 2

Given side b, angle β, and another side a, with side b (opposite β) smaller than a, there are three scenarios that can occur:

1. Angle α (opposite a) = 90 degrees, in which case there would be one possible triangle
2. Angle α is not 90 degrees and there are two possible triangles (you can visualize this by drawing it out and "flipping" side b over the 90 degree projection from corner γ to side c)
3. Side b is too so short, that it can't "reach" both other sides to make a triangle. A way to visualize this is to "attach" b to one end of a, c to the other end, and no matter how you rotate γ or what length you make side c, b is so short that b and c can never connect!

In this case, a quick way to check is to use the Law of Sines, which applies to triangles.

sin(α)/a = sin(β)/b

sin(α) = 46*sin(38°)/27 = 1.049

sin(α) > 1, which is not a real number. Therefore, this isn't a triangle!

So your answer is zero.

Understanding how to visualize these scenarios will allow you to solve any problem like this. Try drawing them out. Note that it doesn't matter which side you make a, b, and c. You will get the same triangle either way, just oriented differently.