Determine whether the given information resutls in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.
b = 4
c = 6
beta = 20
Ok...
I checked my answers in the back of the book and got the first triangle right
Gamma = 30.9
alpha = 129.1
a = 9.1
this is the law of sine stuff
ok also my teacher says that if a second triangle exists...
beta two would be equal to 180 - beta one because it would be an isocoles triangle so
180 - 20 gives me 160
alpha is the same which is 129.1 which is correct according to the back of the book
which runs into a problem because 160 + 129.1 gives me more than 180 degrees
for the second triangle the back of the book says
Gamma 2 = 149.1
alpha 2 = 10.9
a2 = 2.20
aparently then
beta 2 is equal to what?
it's not given in the back
b is not in the book but it would be the same as the first triangle... 4
and I thought a would be 9,1 because it would be the same as the first triangle but apparently i'm wrong
also for some reason c is not given in the back of the book...
So I don't get this what am I not getting here???
My teacher told me in lecture that in problems like this that side a would be the same in both triangles and there would be an issoceles angle and so with two angle betas that would be equal to each other and beta two could be found by taking 180 - beta one sense they are suplementary so I don't get it...
You will find lots of information here,
the first page is not bad
http://www.google.ca/#hl=en&source=hp&q=solving+triangles%2C+the+ambiguous+case&btnG=Google+Search&meta=&aq=f&oq=solving+triangles%2C+the+ambiguous+case&fp=b664983305b547ce
To determine whether the given information results in one triangle, two triangles, or no triangle at all, we can use the Law of Sines. According to the Law of Sines, the ratio of the length of a side of a triangle to the sine of the opposite angle is constant.
Let's calculate the values for the first triangle using the given information:
b = 4
c = 6
β = 20 degrees
Using the Law of Sines:
a/sin(α) = b/sin(β) = c/sin(γ)
a/sin(α) = 4/sin(20) --> (1)
6/sin(γ) = 4/sin(20) --> (2)
To find α and γ, we can use the fact that the sum of the angles in a triangle is 180 degrees:
α + β + γ = 180
β = 20 degrees (given)
Rearranging the equation:
α + γ = 180 - β
α + γ = 180 - 20
α + γ = 160 degrees --> (3)
Now, solving equations (1), (2), and (3) simultaneously will give us the values of α and γ.
Solving equation (1) for a:
a = (4 * sin(α)) / sin(20)
Substituting this value in equation (2):
6/sin(γ) = (4 * sin(α)) / sin(20)
Now, substitute the value of α + γ from equation (3):
6 / sin(γ) = (4 * sin(α)) / sin(20)
6 / sin(γ) = (4 * sin(160 - α)) / sin(20)
Now, you can solve this equation to find the value of γ. Similarly, you can find the value of α.
Finding the second triangle:
You correctly identified that γ2 = α1 since the triangle is isosceles. However, you made a mistake when determining β2. The sum of the angles in a triangle is always 180 degrees, so β2 = 180 - α2 - γ2.
Since you found the values of α and γ for the first triangle, you can substitute those values into the equation to find β2:
β2 = 180 - α1 - γ1
Using this equation, calculate the value of β2.
For the second triangle, you can use the same values of b, c, and β as the first triangle, since these values don't change. However, since the triangle is different, the remaining side and angles will be different. You will need to solve the triangle again using the Law of Sines, this time using the new angles you found.
Remember that the values in the back of the book are not always complete or accurate, so it's important to understand the concepts and calculations yourself.