Question

Questions 8 - 10, deal with the ambiguous SSA case. For each, find all possible solutions and sketch the triangle(s) in each case.

Answer 1

for SSA we have two possibilities depending on the arrangement of the side across from the angle

Answer 2

so first you'd use law of sines to find angle B using the left triangle, solve for the remaining sides/angles

Answer 3

as a reminder:

a/sin(A) = b/sin(B) for law of sines

Answer 4

15/sin(50) = 12/sin(B)

Answer 5

awesome, now solve for B

Answer 6

0.66
2.48

Answer 7

DO i round to the nearest tenth or hundredth?

Answer 8

you should only be getting 1 value for B

also the solution should be in degrees not radians since the angle you are given is also in degrees

try cross multiplying

15/sin(50) = 12/sin(B)

and solving for sin(B), then take the arcsin of the result

Answer 9

4 sin (50)/5

Answer 10

good, then take the arcsin of 4 sin (50)/5

Answer 11

0.61

Answer 12

i think :/

Answer 13

wait 39.79

Answer 14

remember your answer should be in degrees not radians

Answer 15

good, 37.79 degrees for angle B

now you have angle A and angle B, find angle C

Answer 16

i use the c^2=a^2+b^2-2ab cosC right

Answer 17

hint: all the triangles add up to 180 degrees

you can't use law of cosines yet because you don't have c

Answer 18

*all the angles in a triangle

Answer 19

Im not sure i I would use the sin (a)-a formula or just subtract 180-

Answer 20

a/sin(A) = b/sin(B)

Answer 21

angle A + angle B + angle C = 180

so angle C = ?

Answer 22

something like this right

Answer 23

oh okay

180-37.79-50=92.21

Answer 24

for angle c

Answer 25

good

now you have angle C, you can now use law of cosines to find c

Answer 26

okay

c^2=a^2+b^2-2ab (cos(C))
c^2=15^2+12^2-2*15*12(cos(C)

Answer 27

like this

Answer 28

good but you need to plug in angle C = 92.21

Answer 29

oh right..

Answer 30

c^2=15^2+12^2-2*15*12(cos(92.21)
c^2=531.07
c=24.05

Answer 31

wait a min

Answer 32

sure.

Answer 33

awesome, so that's the first ambiguous case solved

however, for the SSA case, angle B can also be obtuse [see the diagram]

so we take 180 - the old angle B to get the new angle B

so 180 - 92.21 = 87.79 as the new angle C

then we re-solve the triangle for angle B and side c

Answer 34

so if angle A = 50 and angle C = 87.79 what is the new angle B?

Answer 35

we use law of cosines right

Answer 36

wait nvm

its

180-50-87.79=42.21

Answer 37

yeah i just double checked and apparently there's only one case for this triangle so it's just the other set of solutions (angle B = 37.79, angle C = 92.2, c = 24.05)

Answer 38

so yeah that's #8 done, repeat the process for #9 and start by finding angle B using law of sines

Answer 39

okay so is just the orgiinal ones we found right

Answer 40

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Vocaloid
awesome, so that's the first ambiguous case solved

however, for the SSA case, angle B can also be obtuse [see the diagram]

so we take 180 - the old angle B to get the new angle B

so 180 - 92.21 = 87.79 as the new angle C

then we re-solve the triangle for angle B and side c
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 41

not this

Answer 42

yeah not that

Answer 43

a/sin(A) = b/sin(B)
15/sin(50) = 12/sin(B)
37.79 degrees for angle B
180-37.79-50=92.21 (Angle C)
c^2=15^2+12^2-2*15*12(cos(92.21)
c^2=531.07
c=24.05

Answer 44

so just this

Answer 45

okay we can continue now

Answer 46

yes

Answer 47

a/sin(A) = b/sin(B)
10/sin(50) = 15/sin(B)

Answer 48

3sin(50)/2

Answer 49

\[\frac{ 3 }{ 2 } \cos(\frac{ 2\pi }{ 9})\]

Answer 50

72.64 degrees

Answer 51

so you got

sin(B) = 3sin(50)/2

you just need to take the arcsin of both sides to find what B is, converting to cos isn't helpful here

Answer 52

\(\color{#0cbb34}{\text{Originally Posted by}}\) @zarkam21
72.64 degrees
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 53

is this it

Answer 54

huh

for some reason it's giving me B = 90 degrees

Answer 55

its prob me thats wrong

Answer 56

oh duh

since 3sin(50)/2 is greater than 1 then there's no solution for the arcsin making this triangle have no real solutions

Answer 57

a/sin(A) = b/sin(B)
10/sin(50) = 15/sin(B)
72.64 degrees for angle B

180-72.64-50=57.36(Angle C)
c^2=10^2+15^2-2*10*15(cos(57.36)
c^2=118.44
c=10.88

Answer 58

so this is all not necessary?

Answer 59

yeah, angle B has no solutions meaning this triangle cannot actually exist

moving on to #10 i guess, we'd solve for angle B as usual using law of sines

Answer 60

a/sin(A) = b/sin(B)
1.6/sin(50) = 2/sin(B)
=-6.1

Answer 61

6.28

Answer 62

the angle is 47 not 50

Answer 63

arcsin(2sin(47)/1.6) = ?

Answer 64

1.15

Answer 65

good but what is that in degrees not radians?

Answer 66

69.89

Answer 67

a little off, 66.09 degrees

now find angle C, given angle A = 47 and angle B = 66.09

Answer 68

180-66.09-47=66.91 (Angle C)
c^2=1.6^2+2^2-2*1.6*2(cos(66.91)
c^2=10.35
c=3.22

Answer 69

yeah that's good

Answer 70

I was double checking w/ law of sines and for some reason I get a diff. value for c hm..

Answer 71

oh really

Answer 72

so is the final answer okay or just i recheck

Answer 73

should*

Answer 74

ok apparently you can't use the law of cosines for an ambigious case, in that case we would simply use law of sines to find c

c/sin(C) = a/sin(A)

so

c/sin(66.91) = 1.6/sin(47) gives the value of c

Answer 75

that means we need to re-do the value of c for triangle #8 but we can get back to that after we finish #10 i guess

Answer 76

c/sin(66.91) = 1.6/sin(47)

multiplying both sides by sin(66.91) gives us

c = 1.6 * sin(66.91) / sin(47) = ?

Answer 77

2.01

Answer 78

this would be #10 right

Answer 79

good but for this triangle we have another possible solution

first we take 180 - angle B to get 180 - 66.09 = 113.91 = the other possible angle B value

so for this new case, we have

angle A = 47, angle B = 113.91, angle C = ?

Answer 80

180-113.91-47=19.09

Answer 81

awesome, then use law of sines again to find c

c/sinC = a/sinA, plug in angles C and A, plus side a to solve for side c

Answer 82

c/sinC = a/sinA
c/sin(19.09)/1.6/sin(47)

Answer 83

that / in the middle should be an equal sign

c/sin(19.09) = 1.6/sin(47)

now find c

Answer 84

2.19

Answer 85

0.72

Answer 86

*

Answer 87

good, 0.72 = c giving us the other solution for the ambiguous case

anyway, going back to #8, we found that angle C = 92.21 giving us

c/sin(92.21) = 15/sin(50) giving us c = 19.57

anyway, let's do a quick recap of our solutions:

Answer 88

#8: only one solution

angle B = 37.79 degrees angle C = 92.21 degrees, c = 19.57

#9: no sol'ns (the arcsin of angle B is greater than 1 so no possible angle B values)

#10: two solutions

solution 1: angle B = 66.09 degrees, angle C = 66.91 degrees, c = 2.01

solution 2: angle B = 113.91 degrees, angle C = 19.09 degrees, c = 0.72

Answer 89

and that's it

Answer 90

oh for #8, we got c^2=531.07
c=24.05

Answer 91

its 19.57 instead?

Answer 92

yeah, I realized we can't use law of cosines we have to use law of sines which gives us 19.57