Respuesta :
Answer:
94°
Step-by-step explanation:
Oblique triangles are of two types: acute or obtuse triangle. They are not right triangles since they have no right angle.
This question can be answer using the Law of cosines, which states that:
[tex] \\ c^2 = a^2 + b^2 - 2a*b * cos(\gamma) [/tex] [1]
Where [tex] \\ \gamma [/tex] is the angle of the respective opposite side c, a and b are the other triangles' sides.
Well, since we have all the information needed to find [tex] \\ \gamma [/tex], that is, a = 54, b = 68 and c = 90, we calculate it using the formula [1] as follows:
[tex] \\ 90^2 = 54^2 + 68^2 - 2*(54)*(68)*cos(\gamma) [/tex]
Solving this equation for [tex] \\ cos(\gamma) [/tex]:
[tex] \\ 90^2 - 54^2 - 68^2 = - 2*(54)*(68)*cos(\gamma) [/tex]
[tex] \\ 8100 - 2916 - 4624 = - 7344*cos(\gamma) [/tex]
[tex] \\ 560 = - 7344*cos(\gamma) [/tex]
[tex] \\ \frac{560}{-7344} = cos(\gamma) [/tex]
[tex] \\ cos(\gamma) = - 0.07625 [/tex] (for four significant figures)
To obtain [tex] \\ \gamma [/tex], we need to determine which angle has a [tex] \\ cos(\gamma) = -0.07625. [/tex] For this, it is necessary the [tex] \\ arccos(x) [/tex] function, also expressed as [tex] \\ cos^{-1}(x). [/tex]
Then
[tex] \\ cos^{-1}[cos(\gamma)] = cos^{-1}(-0.07625) [/tex]
[tex] \\ \gamma = cos^{-1}(-0.07625) = 94.37 [/tex] degrees.
Rounding the result to the nearest whole number, the answer is [tex] \\ \gamma = [/tex] 94°.
The rest of the angles can be obtained using the Law of sines.
