Proving Triangle Angles Equation: cos^2 A + cos^2 B + cos^2 C + 2*cosA*cosB*cosC

In summary, the "Proving Triangle Angles Equation" is a mathematical formula derived from the Law of Cosines and the Pythagorean theorem, which states that the sum of the squares of the cosines of the angles in a triangle is equal to 1. This equation is important in mathematics as it helps establish the relationship between triangle angles and provides a fundamental understanding of trigonometry. Each term in the equation represents the squares of the cosine values of the angles, and the equation is widely used in fields such as engineering, physics, and astronomy for real-world applications. Additionally, there are other important trigonometric equations such as the Law of Sines and the Law of Tangents that are related to the "Proving Triangle Ang
  • #1
ER901
2
0

Homework Statement


I need to prove the following equation:
cos^2 A + cos^2 B + cos^2 C + 2*cosA*cosB*cosC = 1
where A,B,C are the angles of a triangle

Homework Equations


A+B+C = 180


The Attempt at a Solution


I substituted the sum of A,B,C into the equation, and now I have something like
... + cos^2(180-A-C)+... so on, but I don't know what to do next.

Thanks
 
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  • #2
Welcome to Physicsforums ER901!

How about using the Cosine rule to express the Cosines as functions of the sides? Theres some messy algebra but it should work out fine.
 
  • #3
for your question! Let's start by breaking down the equation and understanding what each term represents. The left side of the equation is a sum of three cosine squared terms and a product of three cosine terms. The right side is simply the number 1.

To prove this equation, we will use the fact that the sum of the angles in a triangle is always 180 degrees. This is represented by the equation A+B+C=180.

First, let's rewrite the equation in terms of A, B, and C using the substitution A=180-B-C. This gives us:

cos^2(180-B-C) + cos^2(B) + cos^2(C) + 2*cos(180-B-C)*cos(B)*cos(C) = 1

Next, we can use the identity cos(180-x)=-cos(x) to rewrite the first term as:

cos^2(B+C) + cos^2(B) + cos^2(C) - 2*cos(B+C)*cos(B)*cos(C) = 1

Now, let's use the double angle identity cos(2x)=2cos^2(x)-1 to rewrite the first and third terms as:

(2*cos^2(B)*cos^2(C) - 1) + cos^2(B) + cos^2(C) - (2*cos(B)*cos(C)*cos(B+C)) = 1

Simplifying this, we get:

2*cos^2(B)*cos^2(C) + cos^2(B) + cos^2(C) - 2*cos(B)*cos(C)*cos(B+C) = 2

Using the identity cos(x+y)=cos(x)*cos(y)-sin(x)*sin(y), we can rewrite the last term as:

2*cos^2(B)*cos^2(C) + cos^2(B) + cos^2(C) - 2*(cos(B)*cos(B)*cos(C) - sin(B)*sin(C)*cos(B)*cos(C)) = 2

Simplifying this, we get:

2*cos^2(B)*cos^2(C) + cos^2(B) + cos^2(C) - 2*cos^2(B)*cos^2(C) + 2*sin(B)*sin(C)*cos(B)*cos(C) = 2

Now, we can use the Pythagorean identity sin^2(x)+cos^2(x)=
 

Related to Proving Triangle Angles Equation: cos^2 A + cos^2 B + cos^2 C + 2*cosA*cosB*cosC

1. What is the "Proving Triangle Angles Equation" and why is it important in mathematics?

The "Proving Triangle Angles Equation" is a mathematical formula used to prove that the sum of the squares of the cosines of the angles in a triangle is equal to 1. This equation is important because it helps to establish the relationship between the angles in a triangle and provides a fundamental understanding of trigonometry.

2. How is the "Proving Triangle Angles Equation" derived?

The equation can be derived using the Law of Cosines and the Pythagorean theorem. By substituting these formulas into the equation, it can be simplified and proven to be equal to 1.

3. What does each term in the equation represent?

The cos^2 A, cos^2 B, and cos^2 C terms represent the squares of the cosine values of each angle in the triangle. The 2*cosA*cosB*cosC term represents the product of the cosine values of all three angles in the triangle.

4. How is the "Proving Triangle Angles Equation" used in real-world applications?

The equation is used in various fields such as engineering, physics, and astronomy to calculate and understand the relationships between angles in a triangle. It is also used in navigation and surveying to determine distances and angles between points.

5. Are there any other important trigonometric equations related to the "Proving Triangle Angles Equation"?

Yes, there are several other trigonometric equations that are related to the "Proving Triangle Angles Equation", such as the Law of Sines and the Law of Tangents. These equations are used to solve various problems involving triangles and their angles.

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