Question

Two sides and an angle are given below. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). b=4,c=5,B=40 ∘

Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Type an integer or decimal rounded to two decimal places as needed.)
A. Asingle triangle is produced, where C≈_____,A≈_____, and a ≈ _________
B. Two triangles are produced, where the triangle with the smaller angle C has C₁ ≈_________ ,A₁​ ≈______ a₁ ≈ ________and the triangle with the larger angle C has C₂ ≈________,A₂ ≈________, and a₂ ≈__________
C. No triangles are produced.

Answer 1

To determine if there is a triangle and how many, use the Law of Sines with given sides and angle. Calculate if sin(C) is valid, determine angle C and angle A, then find side a. The solution will reveal if there is one, two, or no triangles possible.

Step-by-step explanation:

To determine whether the given information results in a triangle, and if so, how many, we must assess the given sides and angle using the Law of Sines. The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides and angles. Using the given information, b=4, c=5, and angle B=40°, we first try to find angle C using the Law of Sines:

1. sin(B) / b = sin(C) / c

2. sin(40°) / 4 = sin(C) / 5

3. sin(C) = 5 · sin(40°) / 4

4. Calculate the numerical value of sin(C).

If sin(C) is greater than 1, no such triangle exists. If sin(C) is less than 1 but greater than 0, there may be one or two possible triangles. In this case:

• sin(C) = 5 · sin(40°) / 4

• Evaluate the sin(40°) to determine the value of sin(C).

• If sin(C) <= 1, calculate the angle C using inverse sine, leading to potential angles C or 180-C (if sin(C) results in two possible angles).

• Finally, use angle-sum of triangles (180°) to find angle A, and use Law of Sines again to solve for side a.

Based on the calculated values, we will know if the situation results in one triangle, two triangles, or no triangle. The exact values would need to be computed to fill in the blanks regarding angles A and C, as well as side a