Pythagorean Identity

Definition

The Pythagorean identity is a fundamental trigonometric identity that relates the trigonometric functions sine, cosine, and tangent. It is a crucial concept in understanding the unit circle and verifying, simplifying, and solving trigonometric expressions and 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 always equal to 1: $\sin^2 x + \cos^2 x = 1$.
2. This identity is derived from the properties of the unit circle, where the coordinates of a point on the circle are the sine and cosine of the angle.
3. The Pythagorean identity is used to verify and simplify trigonometric identities, as well as to solve trigonometric equations.
4. The Pythagorean identity can be extended to other trigonometric functions, such as $\tan^2 x + 1 = \sec^2 x$.
5. Understanding the Pythagorean identity is crucial for working with the unit circle, trigonometric identities, and solving trigonometric equations.

Review Questions

• Explain how the Pythagorean identity is derived from the properties of the unit circle.
  • The Pythagorean identity $\sin^2 x + \cos^2 x = 1$ is derived from the properties of the unit circle. In the unit circle, the coordinates of a point on the circle are the sine and cosine of the angle. Since the radius of the unit circle is 1, the sum of the squares of the sine and cosine functions must be equal to 1, as per the Pythagorean theorem. This relationship is the foundation of the Pythagorean identity and is essential for understanding the trigonometric functions and their relationships within the unit circle.
• Describe how the Pythagorean identity can be used to verify and simplify trigonometric identities.
  • The Pythagorean identity $\sin^2 x + \cos^2 x = 1$ can be used to verify and simplify trigonometric identities. For example, when verifying the identity $\sec^2 x = 1 + \tan^2 x$, we can use the Pythagorean identity to substitute $\sin^2 x + \cos^2 x = 1$ and then rearrange the terms to obtain the desired identity. Similarly, the Pythagorean identity can be used to simplify trigonometric expressions by substituting the appropriate trigonometric functions, which can lead to more concise and manageable forms.
• Explain how the Pythagorean identity is used to solve trigonometric equations.
  • The Pythagorean identity $\sin^2 x + \cos^2 x = 1$ is essential for solving trigonometric equations. When faced with an equation involving trigonometric functions, such as $\sin^2 x + \cos^2 x = k$, where $k$ is a constant, the Pythagorean identity can be used to isolate and solve for the unknown variable $x$. By recognizing that the left-hand side of the equation must equal 1 due to the Pythagorean identity, the equation can be simplified and rearranged to find the values of $x$ that satisfy the original equation.

Related terms

Unit Circle: A circular representation of the trigonometric functions, where the coordinates of a point on the circle correspond to the values of sine, cosine, and tangent.

Trigonometric Identities: Equalities involving trigonometric functions that are true for all values of the variables, used to simplify and verify trigonometric expressions.

Trigonometric Equations: Equations involving trigonometric functions that need to be solved to find the values of the variables that satisfy the equation.