Pythagorean Identity

Definition

The Pythagorean identity is a fundamental trigonometric identity that relates the values of the sine, cosine, and tangent functions. It is a crucial concept in understanding the behavior and properties of trigonometric functions, particularly in the context of the unit circle, sum and difference identities, double-angle and half-angle formulas, and solving trigonometric equations.

5 Must Know Facts For Your Next Test

1. The Pythagorean identity states that the sum of the squares of the sine and cosine functions is equal to 1: $\sin^2(x) + \cos^2(x) = 1$.
2. This identity is derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
3. The Pythagorean identity is fundamental to understanding the behavior of trigonometric functions and their relationships within the unit circle.
4. The Pythagorean identity is used to simplify and manipulate trigonometric expressions, particularly in the context of sum and difference identities, double-angle and half-angle formulas, and solving trigonometric equations.
5. The Pythagorean identity is also used to derive other important trigonometric identities, such as the tangent identity: $\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}$.

Review Questions

• Explain how the Pythagorean identity is derived from the Pythagorean theorem and how it relates to the unit circle.
  • The Pythagorean identity, $\sin^2(x) + \cos^2(x) = 1$, is derived directly from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In the context of the unit circle, the sine and cosine functions represent the $y$-coordinate and $x$-coordinate, respectively, of a point on the circle. The Pythagorean identity reflects the fact that the distance from the origin to any point on the unit circle is always 1 unit, which is the radius of the circle. This relationship is fundamental to understanding the behavior and properties of trigonometric functions within the unit circle.
• Describe how the Pythagorean identity is used in the derivation of other important trigonometric identities, such as the tangent identity.
  • The Pythagorean identity is used as a building block to derive other important trigonometric identities, such as the tangent identity: $\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}$. To derive the tangent identity, one can start with the Pythagorean identity, $\sin^2(x) + \cos^2(x) = 1$, and then rearrange the terms to isolate $\sin^2(x)$ and $\cos^2(x)$. Substituting these expressions into the definition of the tangent function, $\tan(x) = \frac{\sin(x)}{\cos(x)}$, leads to the tangent identity. This demonstrates how the Pythagorean identity serves as a fundamental relationship that enables the derivation of other useful trigonometric identities.
• Analyze how the Pythagorean identity is applied in the context of solving trigonometric equations and simplifying trigonometric expressions.
  • The Pythagorean identity is a crucial tool in solving trigonometric equations and simplifying trigonometric expressions. When faced with equations or expressions involving sine and cosine functions, the Pythagorean identity can be used to eliminate one of the functions in favor of the other, reducing the complexity of the problem. For example, when solving a trigonometric equation of the form $\sin^2(x) + \cos^2(x) = k$, where $k$ is a constant, the Pythagorean identity can be applied to simplify the equation and find the possible values of $x$. Similarly, in simplifying trigonometric expressions, the Pythagorean identity can be used to rewrite expressions in terms of a single trigonometric function, making the manipulation and evaluation of the expression more straightforward. The versatility of the Pythagorean identity in these contexts demonstrates its fundamental importance in the study of trigonometry.

Related terms

Unit Circle: A circle with a radius of 1 unit, centered at the origin of a coordinate plane, used to define and relate the trigonometric functions.

Trigonometric Identities: Equations that are true for all values of the variables, used to simplify and manipulate trigonometric expressions.

Trigonometric Functions: Functions that relate the sides and angles of a right triangle, including sine, cosine, and tangent.