Law of Sine and Law of Cosines

Question

Assume 𝛼 is opposite side a, 𝛽 is opposite side b, and 𝛾 is opposite side c. Solve the triangle, if possible. Round the answer to the nearest tenth. (If not possible, enter IMPOSSIBLE. Below, enter your answers so that 𝛽₁ is smaller than 𝛽₂.)

𝛼 = 43.7°,    a = 184.3,    b = 242.8

Find

B1 B2

Y1 Y2

C1 C2

Answer 1

draw a line length=242.8 stating at origin and on positive x-axis.

the origin is point C of the triangle and at end of line is point A of the triangle.

at point A, measure 43.5 degrees clockwise from line and draw in arbitrarily long side c

slanting back toward point C. side a=184.3 eminating from point C is rotatable.

rotate side a to see it meets side c in two spots. when angle C is 21.7 degrees and when angle C=72 degrees.

Answer 2

sin43.7/184.3 = sinB/242.8

sinB = 242.8sin43.7/184.3

=242.3(.690882411)/184.3

=167.4008/184.3

= .90831

B = about 65.27 degrees or rounded to one decimal

B = 65.27 degrees

Y = 180-65.27-43.7= 116-43.7 = 71.03 degrees

Y= 71.03 degrees

c/sin71.03 = 242.8/sin65.27

c = 242.8sin71.03/.90829 = 252.80

c =252.80

It helps to draw the triangle

one is acute (all angles less than 90 degrees), the other obtuse (one angle>90)

the obtuse triangle is inscribed inside the acute triangle with just the position of side a different, making it a smaller triangle. a slants downward in the acute triangle, but upward in the obtuse triangle creating 2 different B angles and 2 different sides b. a, c, and A remain the same, the other angles and dimensions come in two different values: B, Y, and b

for the acute triangle angles are

43.7, 65.3 and 71 opposite sides are 184.3, 242.8 and 252.8

for the obtuse triangle, angles are

27.3 - 43.7 - 109 degrees with opposite sides 122.48 - 184.3 -252.8

b/sinB = c/sinC

242.8/sin65.27 = 252.8/sinC

sinC = 252.8sin65.27/242.8

sinC =252.8(.90829)/242.8

sinC =.9457

C =71.03 or 180-71.03 = 108.97

C = 108.97 for the obtuse triangle

b/sinB = a/sinA

b = asinB/sinA

b =184.3sin27.3/sin43.7

b=84.62/sin43.7

b = 122.48

B1 B2 27.3, 65.3

Y1 Y2 109, 71

c1 c2 252.8, 252.8