1) cos α = 0.8, cos β = 0.6, cos γ = 0. The triangle exists and is right.
2) cos α ≈ 0.882, cos β ≈ 0.471, cos γ ≈ 0.441. The triangle exists and is acute.
3) cos α ≈ 0.977, cos β ≈ 0.324, cos γ ≈ -0.115. The triangle exists and is obtuse.
4) cos α = 1, cos β = 1, cos γ = -1. The triangle does not exist.
How to determine if a triangle may exist and the existence of given angles
Triangles are geometric figures formed by three line segments and internal angles, whose sum is equal to 180°, which can be three acute angles or an obtuse angle and two acute angles or a right angle and two acute angles. The possibility is determined by applying the law of cosine:
[tex]\cos \theta = -\frac{x^{2}-y^{2}-z^{2}}{2\cdot y\cdot z}[/tex] (1)
The triangle if and only if the cosine of each angle is between -1 and 1 with the following characteristics:
- If the cosine of the angle is between 0 and 1, then it is acute.
- If the cosine of the angle is 0, then it is right.
- If the cosine of the angle is between 0 and -1, then it is obtuse.
Now we proceed to determine the existence of each triangle:
1) cos α = 0.8, cos β = 0.6, cos γ = 0. The triangle exists and is right.
2) cos α ≈ 0.882, cos β ≈ 0.471, cos γ ≈ 0.441. The triangle exists and is acute.
3) cos α ≈ 0.977, cos β ≈ 0.324, cos γ ≈ -0.115. The triangle exists and is obtuse.
4) cos α = 1, cos β = 1, cos γ = -1. The triangle does not exist.
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