Law of Sines and Cosines

[Chaim]

C----35----A
--\---------/
---\------/
---28--17
-----\--/
------\/
------B
This is what the triangle is suppose to be like.
Law of sine: (sinA/a)=(sinB/b)=(sinC/c)
Law of cosines: a²=b²+c²-2bc(cosA)
b²=a²+c²-2ac(cosB)
c²=a²+b²-2ab(cosC)
Capital letters are the angles, while lowercase letters are the side lengths
a=28, b=35, c=17
Basically I'm trying to find all the angles of the triangle.

First I find angle A
28²=35²+17²-2(35)(17)cosA
784=1514-1190cosA
-730=1190cosA
Angle A is about 52.193 degrees

Then I find Angle B, but this is where I'm confused at, I try to find it 3 different ways:
1. Using Law of Sine
(sin52.193/28)=(sinB/35)
sinB = 0.988
sin^-1(0.988)=81.115 degrees=Angle B

2. Using Law of Sine
(sin52.193/28)=(sinC/17)
sinC = 0.479
sin^-1(0.479)=28.620
180-52.193-28.620=99.187=Angle B

3. Using Law of Cosine
35²=28²+17²-2(28)(27)(cosB)
1225=1073-2(952cosB)
152=1904cosB
0.0799=cosB
cos^-1(0.0799)=85.4162=Angle B

So basically my teacher got somewhere around 99 degrees for angle B, which is close to what I got for solution #2
But I was wondering, how do you find out which one to use, because they all seem to work
Thanks!

 

[Deleted member 4993]

C----35----A
--\---------/
---\------/
---28--17
-----\--/
------\/
------B
This is what the triangle is suppose to be like.
Law of sine: (sinA/a)=(sinB/b)=(sinC/c)
Law of cosines: a²=b²+c²-2bc(cosA)
b²=a²+c²-2ac(cosB)
c²=a²+b²-2ab(cosC)
Capital letters are the angles, while lowercase letters are the side lengths
a=28, b=35, c=17
Basically I'm trying to find all the angles of the triangle.

First I find angle A
28²=35²+17²-2(35)(17)cosA
784=1514-1190cosA
-730=1190cosA
Angle A is about 52.193 degrees

Then I find Angle B, but this is where I'm confused at, I try to find it 3 different ways:
1. Using Law of Sine
(sin52.193/28)=(sinB/35)
sinB = 0.988
sin^-1(0.988)=81.115 degrees=Angle B

2. Using Law of Sine
(sin52.193/28)=(sinC/17)
sinC = 0.479
sin^-1(0.479)=28.620
180-52.193-28.620=99.187=Angle B

3. Using Law of Cosine
35²=28²+17²-2(28)(27)(cosB)
1225=1073-2(952cosB)
152=1904cosB
0.0799=cosB
cos^-1(0.0799)=85.4162=Angle B

So basically my teacher got somewhere around 99 degrees for angle B, which is close to what I got for solution #2
But I was wondering, how do you find out which one to use, because they all seem to work
Thanks!

While calculating sin^-1, you must remember:

sin(Θ) = sin(180°-Θ)

So you need to know what the triangle looks like before deciding on the angle. Cosine does not have that problem.....

 

[biffboy]

C----35----A
--\---------/
---\------/
---28--17
-----\--/
------\/
------B
This is what the triangle is suppose to be like.
Law of sine: (sinA/a)=(sinB/b)=(sinC/c)
Law of cosines: a²=b²+c²-2bc(cosA)
b²=a²+c²-2ac(cosB)
c²=a²+b²-2ab(cosC)
Capital letters are the angles, while lowercase letters are the side lengths
a=28, b=35, c=17
Basically I'm trying to find all the angles of the triangle.

First I find angle A
28²=35²+17²-2(35)(17)cosA
784=1514-1190cosA
-730=1190cosA
Angle A is about 52.193 degrees

Then I find Angle B, but this is where I'm confused at, I try to find it 3 different ways:
1. Using Law of Sine
(sin52.193/28)=(sinB/35)
sinB = 0.988
sin^-1(0.988)=81.115 degrees=Angle B

2. Using Law of Sine
(sin52.193/28)=(sinC/17)
sinC = 0.479
sin^-1(0.479)=28.620
180-52.193-28.620=99.187=Angle B

3. Using Law of Cosine
35²=28²+17²-2(28)(27)(cosB)
1225=1073-2(952cosB)
152=1904cosB
0.0799=cosB
cos^-1(0.0799)=85.4162=Angle B

So basically my teacher got somewhere around 99 degrees for angle B, which is close to what I got for solution #2
But I was wondering, how do you find out which one to use, because they all seem to work
Thanks!

You have an error in 3 when using cosine rule to get angle B You should have 1225=1073-952cosB So cosB= -152/952 So B=99.187
(When using cos rule there is never any ambiguity about the angle)
When using sine rule to get an angle the answer can be what your calculator says or (180 -that)
In this question because 35^2 is greater than (17^2+28^2) I know B will be an obtuse angle. So it is 99.187 as the cos rule gave us.
Also I get A to be 52.16