Assume 𝛼 is opposite side a, 𝛽 is opposite side b, and 𝛾 is opposite side c. Assume that a = 7sqrt(3), b = 7sqrt(2), 𝛽 = 45°. Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Enter your answers so that 𝛼1 is greater than 𝛼2.)
(To the extent possible use standard angles to simplify your responses. When necessary express your answers using the functions sin, cos, sin−1, cos−1. If there is no solution enter IMPOSSIBLE.)
You are looking at what is called the "ambiguous case"
Let's find angle 𝛼 , using the sine law
sin𝛼/7√3 = sin 45/7√2
sin 𝛼 = 7√3(√2/2) / 7√2 = √3/2
so 𝛼 = 60° or 𝛼 = 120° , since the sine is positive in both the 1st and 2nd quadrants.
So it looks like we could have 2 triangles.
Triangle #1, angles B = 45, angle A = 60 and angle C = 75°
find the missing side c using the sine law.
triangle #2, angles 45, 120 and 15° , again find the missing side.
Side c should be the same for both.
sin𝛼/7√3 = sin45°/7√2
now, 𝛾 = 180-𝛼-𝛽
and you can use the law of sines/cosines to find c
To determine the number of triangles and solve them, we can apply the law of sines given by:
sin(𝛼)/a = sin(𝛽)/b = sin(𝛾)/c
Given that a = 7√3, b = 7√2, and 𝛽 = 45°, let's use the law of sines to find the value of angle 𝛼.
sin(𝛽)/b = sin(𝛼)/a
sin(45°)/(7√2) = sin(𝛼)/(7√3)
Simplifying the equation:
1/(√2 * 7√2) = sin(𝛼)/(7√3)
1/(2 * 7) = sin(𝛼)/(7√3)
1/14 = sin(𝛼)/(7√3)
We can solve for sin(𝛼):
sin(𝛼) = (1/14) * (7√3)
sin(𝛼) = √3/14
Now, let's find the value of angle 𝛼 using the arcsin function (sin⁻¹):
𝛼 = sin⁻¹(√3/14)
𝛼 ≈ 20.06°
Since we have determined the value of angle 𝛼, we can check if there are one or two possible triangles.
If 𝛼 > 𝛽, then there is one possible triangle.
If 𝛼 < 𝛽, then there are two possible triangles.
In this case, 𝛼 (20.06°) is less than 𝛽 (45°), so there are two possible triangles.
Now, let's solve each triangle.
Triangle 1:
Given: 𝛼 = 20.06°, 𝛽 = 45°, a = 7√3, b = 7√2
To find side c, we can use the law of sines:
sin(𝛽)/b = sin(𝛾)/c
sin(45°)/(7√2) = sin(𝛾)/c
c * sin(45°) = (7√2) * sin(𝛾)
c = [(7√2) * sin(𝛾)] / sin(45°)
Using sin(𝛾) = sin(180° - 𝛼 - 𝛽):
c = [(7√2) * sin(180° - 20.06° - 45°)] / sin(45°)
c ≈ 4.983
Triangle 2:
Given: 𝛼 = 20.06°, 𝛽 = 45°, a = 7√3, b = 7√2
To find side c, we can use the law of sines:
sin(𝛽)/b = sin(𝛾)/c
sin(45°)/(7√2) = sin(𝛾)/c
c * sin(45°) = (7√2) * sin(𝛾)
c = [(7√2) * sin(𝛾)] / sin(45°)
Using sin(𝛾) = sin(𝛼 + 𝛽):
c = [(7√2) * sin(20.06° + 45°)] / sin(45°)
c ≈ 18.206
Therefore, we have found two possible triangles with the following side lengths:
Triangle 1: 𝛼 = 20.06°, 𝛽 = 45°, 𝛾 ≈ 4.983
Triangle 2: 𝛼 = 20.06°, 𝛽 = 45°, 𝛾 ≈ 18.206
To determine the number of triangles, we can apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we have:
a = 7√3
b = 7√2
β = 45°
To find the possible values for side c (opposite angle 𝛾), we can rearrange the inequality as follows:
a + b > c
7√3 + 7√2 > c
To simplify this inequality, we can compute the actual values:
7√3 ≈ 7 * 1.732 ≈ 12.124
7√2 ≈ 7 * 1.414 ≈ 9.898
So, our inequality becomes:
12.124 + 9.898 > c
21.022 > c
Therefore, we know that side c (opposite angle 𝛾) must be less than 21.022 units in length for there to be a triangle.
Now, let's solve the triangle. We know side b = 7√2 = 9.898 and angle β = 45°.
To find side a (opposite angle α), we can use the law of sines:
sin α / b = sin β / a
Rearranging the equation for side a:
a = b * sin α / sin β
Plugging in the values:
a = 9.898 * sin α / sin 45°
Now, we need to find the value of sin α. Since we know side a and angle α, we can rearrange the equation for the law of sines to solve for sin α:
sin α = a * sin β / b
Plugging in the values:
sin α = (7√3 * sin 45°) / 7√2
sin α = (7 * 1.732 * 0.707) / (7 * 1.414)
sin α ≈ 1.198
However, the sine of an angle cannot be greater than 1. Therefore, there are no possible solutions for angle α and hence no triangle exists.
In summary, the answer is IMPOSSIBLE. There is no triangle possible with the given information.