is there a lowest or least cosine of a triangle with sides of 4,5,and 6?
such as: .78, .72...
To find the lowest or least cosine of a triangle with sides of 4, 5, and 6, we can use the Law of Cosines. The Law of Cosines is an equation that relates the lengths of the sides of a triangle to the cosine of one of its angles.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this equation, c represents the side opposite the angle C, and a and b represent the other two sides of the triangle.
Let's calculate the cosine of each angle in the triangle using the Law of Cosines and see which one is the lowest or least.
For angle A, using the Law of Cosines:
a^2 = b^2 + c^2 - 2bc * cos(A)
4^2 = 5^2 + 6^2 - 2 * 5 * 6 * cos(A)
16 = 25 + 36 - 60 * cos(A)
16 - 61 = -60 * cos(A)
-45 = -60 * cos(A)
cos(A) = -45 / (-60)
cos(A) = 0.75
For angle B, using the Law of Cosines:
b^2 = a^2 + c^2 - 2ac * cos(B)
5^2 = 4^2 + 6^2 - 2 * 4 * 6 * cos(B)
25 = 16 + 36 - 48 * cos(B)
25 - 52 = -48 * cos(B)
-27 = -48 * cos(B)
cos(B) = -27 / (-48)
cos(B) ≈ 0.5625
For angle C, using the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
6^2 = 4^2 + 5^2 - 2 * 4 * 5 * cos(C)
36 = 16 + 25 - 40 * cos(C)
36 - 41 = -40 * cos(C)
-5 = -40 * cos(C)
cos(C) = -5 / (-40)
cos(C) ≈ 0.125
From the calculations, we can see that the least cosine value is approximately 0.125, which belongs to angle C. Therefore, the lowest or least cosine of this triangle is approximately 0.125.