Proving the Relationship between Cosine of Triangle Angles: cos(180° - A) -cosA
When dealing with triangle angles, understanding the relationships between their trigonometric functions can provide valuable insights. One such relationship, which is fundamental to solving a variety of geometry and trigonometry problems, is the identity cos(180° - A) -cosA. This article will guide you through a detailed proof of this identity using the trigonometric identity for the cosine of the sum of two angles.
Understanding the Trigonometric Identity for the Cosine of the Sum of Two Angles
The trigonometric identity for the cosine of the sum of two angles is:
$$cos(A B) cosA cdot cosB - sinA cdot sinB$$This identity is a crucial tool in trigonometry and can be applied to various problems, including proving the relationship involving supplementary angles in a triangle.
Step-by-Step Proof: cos(180° - A) -cosA
Let's start by considering a triangle with angles A, B, and C. According to the Triangle Angle Sum Property, the angles of a triangle add up to 180 degrees:
$$A B C 180°$$Rearranging this equation, we get:
$$B C 180° - A$$Now, let's apply the cosine identity to the angles B and C. Specifically, we are interested in the cosine of the sum B C:
$$cos(B C) cos(180° - A)$$Using the identity for the cosine of the sum of two angles:
$$cos(B C) cosB cdot cosC - sinB cdot sinC$$Given that B C 180° - A, we substitute 180° - A for B C:
$$cos(180° - A) cos(180° - A)$$Now, using a trigonometric identity for the cosine of supplementary angles, we know:
$$cos(180° - x) -cosx$$Substituting A for x in this identity, we get:
$$cos(180° - A) -cosA$$Therefore, we have proven that for any angle A in a triangle, the relationship cos(180° - A) -cosA holds true.
Illustrative Examples
To further illustrate this proof, let's consider a specific example. Suppose we have a triangle with angles A 30° and B 60°. Using the sum of angles in a triangle (A B C 180°), we can find angle C:
$$30° 60° C 180°$$Solving for C, we get:
$$C 90°$$Now, let's find cos(180° - A) and compare it to cosA:
First, calculate cos(180° - 30°):
$$cos(180° - 30°) cos150° -cos30°$$Next, calculate cos30°:
$$cos30° sqrt{3}/2$$Therefore:
$$cos(180° - 30°) -sqrt{3}/2$$Since cos30° sqrt{3}/2, the identity holds true:
$$cos(180° - A) -cosA$$This example demonstrates how the relationship cos(180° - A) -cosA can be verified.
Conclusion
In conclusion, we have proven that for the angles A, B, and C of a triangle, the relationship cos(180° - A) -cosA is always true. This identity is a powerful tool in solving trigonometric and geometric problems, and understanding it can greatly enhance your problem-solving skills in mathematics.