based on pythagorean identities which equation is true

Guri Bee

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21-10-2024

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Unraveling the Mystery: Which Pythagorean Identity is True?

The Pythagorean identities are fundamental to trigonometry, offering a powerful relationship between the sine, cosine, and tangent functions. But which one is true? Let's dive into the world of these identities and find our answer!

The Key Players

Before we tackle the question, let's introduce the three main Pythagorean identities:

1. sin²θ + cos²θ = 1
2. 1 + tan²θ = sec²θ
3. 1 + cot²θ = csc²θ

Understanding the Basics

These identities are derived from the fundamental Pythagorean theorem (a² + b² = c²), applied to a right triangle. Let's break down the first identity (sin²θ + cos²θ = 1) as an example:

• θ: Represents an angle in a right triangle.
• sin θ: The ratio of the opposite side to the hypotenuse.
• cos θ: The ratio of the adjacent side to the hypotenuse.

The Proof

Imagine a right triangle with angle θ. The Pythagorean theorem states that the square of the hypotenuse (c²) is equal to the sum of the squares of the other two sides (a² + b²).

Now, if we divide both sides of the equation by c² (the hypotenuse squared), we get:

(a²/c²) + (b²/c²) = 1

Notice that (a/c) is simply sin θ and (b/c) is cos θ. Substituting these values, we get:

sin²θ + cos²θ = 1

Exploring the Other Identities

The remaining two identities can be similarly derived using the relationships between sine, cosine, and the other trigonometric functions (tangent, secant, cosecant, cotangent).

• Identity 2 (1 + tan²θ = sec²θ): This identity follows from the definition of tangent (sin θ / cos θ) and secant (1/cos θ).
• Identity 3 (1 + cot²θ = csc²θ): This identity is derived using the definitions of cotangent (cos θ / sin θ) and cosecant (1/sin θ).

The Answer

The question asked which equation is true, implying that all three Pythagorean identities are valid. Therefore, all three identities are true.

Practical Applications

These identities are not merely theoretical concepts. They are crucial for solving various trigonometric problems, including:

• Simplifying trigonometric expressions: They help us rewrite expressions involving trigonometric functions in a more manageable form.
• Solving trigonometric equations: They can be used to solve equations involving trigonometric functions.
• Finding unknown angles and sides in right triangles: They play a vital role in solving trigonometric problems involving right triangles.

Conclusion

The Pythagorean identities offer a fundamental understanding of the relationships between trigonometric functions. They are powerful tools used in various branches of mathematics, physics, and engineering. Understanding and applying these identities opens a doorway to a deeper appreciation of the world of trigonometry.

Note: This article incorporates information and examples from various sources, including online resources and textbooks.