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Questions and Answers
What is the Pythagorean identity based on?
What is the Pythagorean identity based on?
What is the result of dividing each term by the square of the side length (b²) in the Pythagorean theorem?
What is the result of dividing each term by the square of the side length (b²) in the Pythagorean theorem?
What is the simplified form of the equation a² / b² + b² / b² = c² / b²?
What is the simplified form of the equation a² / b² + b² / b² = c² / b²?
What is the relation between the trigonometric ratios given by the Pythagorean identity?
What is the relation between the trigonometric ratios given by the Pythagorean identity?
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What is an application of the Pythagorean identity?
What is an application of the Pythagorean identity?
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What is the result of taking the square root of both sides of the equation (a / b)² + 1 = (c / b)²?
What is the result of taking the square root of both sides of the equation (a / b)² + 1 = (c / b)²?
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Study Notes
Pythagorean Identity
Introduction
The Pythagorean identity is a fundamental concept in trigonometry. It is derived from the Pythagorean theorem, which states that for any right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides (the legs). This identity is used to establish relationships between trigonometric ratios, such as sine, cosine, tangent, secant, and their reciprocals.
Derivation of the Pythagorean Identity
The Pythagorean identity is based on the Pythagorean theorem. This theorem can be applied to trigonometric ratios in a right-angled triangle, where the trigonometric ratios are defined. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Dividing each term by the square of the side length (b²) gives:
a² / b² + b² / b² = c² / b²
This can be simplified to:
(a / b)² + 1 = (c / b)²
Taking the square root of both sides, we get:
a / b = ± (c / b)
This is known as the Pythagorean identity, which gives the relation between the trigonometric ratios of sine, cosine, and tangent.
Applications of the Pythagorean Identity
The Pythagorean identity is used to prove other trigonometric identities. For example, to prove the identity sin 4x - cos 4x = sin 2x - cos 2x, we can use the Pythagorean identity to rewrite sin² 4x + cos² 4x as 1. The identity is also used to solve problems related to heights and distances, and to find any trigonometric ratio when another ratio is given.
Conclusion
The Pythagorean identity is a crucial concept in trigonometry, providing a relationship between trigonometric ratios. It is derived from the Pythagorean theorem and has numerous applications in solving trigonometric problems. Understanding and mastering the Pythagorean identity is essential for anyone studying trigonometry.
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Description
Learn about the Pythagorean identity, its derivation from the Pythagorean theorem, and its applications in trigonometry, including solving problems related to heights and distances. Mastering this concept is essential for anyone studying trigonometry. Test your understanding with this quiz!
