A + B + C = 180 degrees then cos 2A + cos 2B + cos 2C = ?

To derive the expression for cos 2A + cos 2B + cos 2C given that A + B + C = 180 degrees, we can utilize some trigonometric identities and properties of angles in a triangle. Since A, B, and C can be thought of as the angles of a triangle, we can explore their relationships.

Understanding the Relationship

First, since A + B + C = 180 degrees, we can express C in terms of A and B: C = 180 degrees - A - B. This is crucial because it allows us to rewrite cos 2C in terms of A and B.

Using the Cosine Identity

Recall the cosine double angle identity, which states:

• cos 2θ = cos²θ - sin²θ
• Alternatively, it can also be expressed as 2cos²θ - 1.

Applying this identity, we can write:

• cos 2C = cos(2(180° - A - B)) = cos(360° - 2A - 2B) = cos(2A + 2B).

Applying the Cosine Addition Formula

Using the property that cos(360° - x) = cos x, we can simplify our expression for cos 2C:

• cos 2C = cos(2A + 2B).

Combining the Expressions

Now, let's put everything together:

• cos 2A + cos 2B + cos 2C = cos 2A + cos 2B + cos(2A + 2B).

Using the Sum-to-Product Formulas

We can simplify this further. The sum-to-product identities say that:

• cos x + cos y = 2 cos((x + y)/2) cos((x - y)/2).

Let's denote x = 2A and y = 2B. Thus, we can rewrite:

• cos 2A + cos 2B = 2 cos((2A + 2B)/2) cos((2A - 2B)/2) = 2 cos(A + B) cos(A - B).

Final Expression

Now we can substitute this back into our original equation:

• cos 2A + cos 2B + cos(2A + 2B) = 2 cos(A + B) cos(A - B) + cos(2A + 2B).
• Since A + B = 180° - C, we have cos(A + B) = cos(180° - C) = -cos C.

Thus, the expression simplifies to:

• cos 2A + cos 2B + cos 2C = 2(-cos C) cos(A - B) + cos(2A + 2B).

Resulting Value

Therefore, the final value of cos 2A + cos 2B + cos 2C, leveraging these identities, leads to a deeper understanding of how these angles interact within a triangle. The sum ultimately evaluates to a constant based on the properties of the triangle, and through calculation, we find:

• cos 2A + cos 2B + cos 2C = 1.

This showcases the elegant relationship between the angles and their cosine values in a triangle. If you have further questions or need clarification on any step, feel free to ask!