If a triangle has sides of lengths a and b, which make a C-degree angle, then the length

of the side opposite C is c, where c2 = a2 + b2 − 2ab cosC. This is the SAS version of the Law of Cosines. Explain the terminology. Derive an equivalent SSS version of the Law of Cosines, which gives the cosine of the angle in terms of the lengths of the three sides. Now use it to find the angles of the triangle whose sides have lengths 4 cm, 5 cm, and 6 cm.

2 a b cos C = a^2 + b^2 - c^2

cos C = [a^2 + b^2 -c^2]/2ab

a = 4
b = 5
c = 6

cos C = [ 16 + 25 - 36]/40
etc

angles of 45,135,45,135

The terminology used in the Law of Cosines refers to different aspects of a triangle:

1. SAS: This stands for "Side-Angle-Side". In the context of the Law of Cosines, it refers to a given triangle with known side lengths (a and b) and the included angle (C). This version of the law allows us to find the length of the third side (c) using the formula c^2 = a^2 + b^2 - 2ab*cos(C).

2. SSS: This stands for "Side-Side-Side". In the context of the Law of Cosines, it refers to a given triangle with known side lengths (a, b, and c). The SSS version of the law allows us to find the cosine of one of the angles using the formula cos(C) = (a^2 + b^2 - c^2) / (2ab).

To derive the SSS version of the Law of Cosines, we start with the SAS version:

c^2 = a^2 + b^2 - 2ab*cos(C)

Rearranging this equation, we have:

2ab*cos(C) = a^2 + b^2 - c^2

Dividing both sides by 2ab, we get:

cos(C) = (a^2 + b^2 - c^2) / (2ab)

This equation represents the SSS version of the Law of Cosines, which allows us to find the cosine of an angle (C) in terms of the lengths of the three sides (a, b, and c).

Now, let's use this SSS version to find the angles of a triangle with side lengths 4 cm, 5 cm, and 6 cm:

Let a = 4 cm, b = 5 cm, and c = 6 cm.

Using the SSS version, we can find the cosine of angle C:

cos(C) = (a^2 + b^2 - c^2) / (2ab)
cos(C) = (4^2 + 5^2 - 6^2) / (2*4*5)
cos(C) = (16 + 25 - 36) / 40
cos(C) = 5 / 40
cos(C) = 1 / 8

To find angle C, we can take the inverse cosine (cos^-1) of 1/8:

C = cos^-1(1/8)
C ≈ 82.83 degrees

Now, we can find angles A and B using the triangle's angle sum property:

A + B + C = 180 degrees

A + B + 82.83 = 180

A + B ≈ 180 - 82.83
A + B ≈ 97.17

Since the sum of the angles in a triangle is 180 degrees, we can approximate angles A and B as equal to 97.17 degrees each.

Therefore, in the triangle with side lengths 4 cm, 5 cm, and 6 cm, the angles (rounded to two decimal places) are approximately: A ≈ 97.17 degrees, B ≈ 97.17 degrees, and C ≈ 82.83 degrees.

The SAS version of the Law of Cosines states that in a triangle with sides of lengths a and b, which make a C-degree angle, the length of the side opposite C (denoted as c) can be calculated using the formula c^2 = a^2 + b^2 - 2ab * cos(C).

Let's derive the SSS version of the Law of Cosines, which provides the cosine of the angle in terms of the lengths of the three sides.

In a triangle with sides of lengths a, b, and c, let α, β, and γ be the angles opposite to these sides, respectively. Applying the Law of Cosines to all three sides, we have:

a^2 = b^2 + c^2 - 2bc * cos(α)
b^2 = c^2 + a^2 - 2ca * cos(β)
c^2 = a^2 + b^2 - 2ab * cos(γ)

Solving these equations for cos(α), cos(β), and cos(γ), we get:

cos(α) = (b^2 + c^2 - a^2) / (2bc)
cos(β) = (c^2 + a^2 - b^2) / (2ca)
cos(γ) = (a^2 + b^2 - c^2) / (2ab)

Now, let's use the SSS version of the Law of Cosines to find the angles of the triangle with side lengths of 4 cm, 5 cm, and 6 cm.

Using the formula above, we can calculate:

cos(α) = (5^2 + 6^2 - 4^2) / (2 * 5 * 6)
cos(β) = (6^2 + 4^2 - 5^2) / (2 * 6 * 4)
cos(γ) = (4^2 + 5^2 - 6^2) / (2 * 4 * 5)

Simplifying these calculations, we have:

cos(α) = (25 + 36 - 16) / 60 = 45 / 60 = 0.75
cos(β) = (36 + 16 - 25) / 48 = 27 / 48 = 0.5625
cos(γ) = (16 + 25 - 36) / 40 = 5 / 40 = 0.125

To find the angles α, β, and γ, we can take the inverse cosine (arccos) of these values:

α = arccos(0.75)
β = arccos(0.5625)
γ = arccos(0.125)

Using a calculator in degree mode or a trigonometric table, we can find the respective angle measures:

α ≈ 41.41 degrees
β ≈ 55.50 degrees
γ ≈ 83.09 degrees