Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that B1 is smaller than B2.)
a = 39, c = 40, angle A = 36°
angle B1 = ° angle B2 = °
angle C1 = ° angle C2 = °
b1 = b2 =
Solve triangle ABC. (If an answer does not exist, enter DNE. Round your answers to one decimal place.)
a = 26, b = 32, c = 27
To solve for all possible triangles that satisfy the given conditions using the Law of Sines, we can use the formula:
sin(A) / a = sin(B) / b = sin(C) / c
From the given information, we have:
a = 39
c = 40
angle A = 36°
Let's solve for angle B and angle C first:
sin(A) / a = sin(B) / b
sin(36°) / 39 = sin(B) / b
b * sin(36°) = 39 * sin(B)
b = (39 * sin(B)) / sin(36°)
Similarly,
sin(A) / a = sin(C) / c
sin(36°) / 39 = sin(C) / 40
40 * sin(36°) = 39 * sin(C)
sin(C) = (40 * sin(36°)) / 39
Now, we can solve for the possible values:
angle B1 = sin^(-1)((39 * sin(36°)) / sin^(-1)(40 * sin(36°)) / 39))
angle B2 = 180° - angle B1
angle C1 = sin^(-1)((40 * sin(36°)) / 39)
angle C2 = 180° - angle C1
b1 = (39 * sin(B1)) / sin(36°)
b2 = (39 * sin(B2)) / sin(36°)
Now, let's calculate the exact values:
angle B1 = sin^(-1)((39 * sin(36°)) / (40 * sin(36°)) = 40.1°
angle B2 = 180° - 40.1° = 139.9°
angle C1 = sin^(-1)((40 * sin(36°)) / 39) = 43.9°
angle C2 = 180° - 43.9° = 136.1°
b1 = (39 * sin(40.1°)) / sin(36°) = 27.3
b2 = (39 * sin(139.9°)) / sin(36°) = 11.9
Therefore, the possible triangles are:
1) angle B1 = 40.1°, angle C1 = 43.9°, b1 = 27.3
2) angle B2 = 139.9°, angle C2 = 136.1°, b2 = 11.9
To solve for all possible triangles that satisfy the given conditions using the Law of Sines, follow these steps:
1. First, write down the given information:
- Side a = 39
- Side c = 40
- Angle A = 36°
2. Use the Law of Sines, which states that the ratios of the lengths of the sides of a triangle are proportional to the sines of their opposite angles. The formula is as follows:
a/sin(A) = b/sin(B) = c/sin(C)
3. Substitute the given values into the formula:
39/sin(36°) = b/sin(B) = 40/sin(C)
4. To find angle B, rearrange the equation to solve for sin(B):
sin(B) = (b*sin(36°))/39
5. Use the inverse sine function (sin^-1) to find the value of B:
B = sin^-1[(b*sin(36°))/39]
6. Similarly, to find angle C, rearrange the equation to solve for sin(C):
sin(C) = (40*sin(36°))/39
7. Use the inverse sine function to find angle C:
C = sin^-1[(40*sin(36°))/39]
Now that we have the values of angles B and C, we can proceed to find the possible values for angle B1, B2, C1, C2, b1, and b2. Since the Law of Sines allows for two possible solutions, we will consider these cases:
Case 1: B1 and C1 are the smaller angles, and B2 and C2 are the larger angles.
8. For B1 and C1:
- B1 = B, using the value of B obtained in step 5.
- C1 = C, using the value of C obtained in step 7.
9. For b1:
Use the Law of Sines again:
b1/sin(B1) = a/sin(A)
b1 = (a * sin(B1)) / sin(A)
10. For b2 and C2:
- B2 = 180° - B1
- C2 = 180° - C1
11. For b2:
Use the Law of Sines:
b2/sin(B2) = a/sin(A)
b2 = (a * sin(B2)) / sin(A)
Case 2: B1 and C1 are the larger angles, and B2 and C2 are the smaller angles.
12. For B1 and C1:
- B1 = 180° - B, using the value of B obtained in step 5.
- C1 = 180° - C, using the value of C obtained in step 7.
13. For b1 and C1:
Use the Law of Sines:
b1/sin(B1) = c1/sin(C1)
b1 = (c1 * sin(B1)) / sin(C1)
14. For b2 and C2:
- B2 = B
- C2 = C
15. For b2:
Use the Law of Sines:
b2/sin(B2) = c2/sin(C2)
b2 = (c2 * sin(B2)) / sin(C2)
Now, substitute the values of B1, B2, C1, C2, b1, and b2 into the given expression. Round the answers to one decimal place:
B1 = °
B2 = °
C1 = °
C2 = °
b1 =
b2 =
Please note that without numerical values for angles B and C, it is not possible to provide specific answers for B1, B2, C1, C2, b1, and b2. Remember to substitute the obtained values for B and C to find the actual measurements.
sin 36/39 = sin C /40
C = sin^-1(40/39)sin 36 = 37
B = 180 - A -C = 107
C could also be in Quadrant 2
180 - 37 = 143
then B is 180 - 36 -143 = 1 deg
sinB/b = sin 36/39 = .0151
so
b = sin B/.0151
b1 = sin 1 /.0151 = 1.16
b2 = sin 107/.0151 = 63.3