key term - Pythagorean Identities

Definition

Pythagorean identities are fundamental trigonometric equalities that relate the sine, cosine, and tangent functions to one another. These identities are based on the Pythagorean theorem and provide a powerful tool for simplifying and manipulating trigonometric expressions.

5 Must Know Facts For Your Next Test

1. The Pythagorean identities are fundamental relationships between the sine, cosine, and tangent functions.
2. The three main Pythagorean identities are: $\sin^2(x) + \cos^2(x) = 1$, $\tan^2(x) + 1 = \sec^2(x)$, and $\cot^2(x) + 1 = \csc^2(x)$.
3. Pythagorean identities are derived from the Pythagorean theorem and the unit circle representation of trigonometric functions.
4. Pythagorean identities are useful for simplifying trigonometric expressions, solving trigonometric equations, and proving other trigonometric identities.
5. The Pythagorean identities hold true for any angle, regardless of the quadrant in which the angle is located.

Review Questions

• Explain how the Pythagorean theorem is related to the Pythagorean identities.
  • The Pythagorean identities are 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. This relationship is expressed in the fundamental Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1$. The other Pythagorean identities, such as $\tan^2(x) + 1 = \sec^2(x)$ and $\cot^2(x) + 1 = \csc^2(x)$, are also based on the Pythagorean theorem and the geometric properties of the unit circle.
• Describe how Pythagorean identities can be used to simplify trigonometric expressions.
  • Pythagorean identities provide a powerful tool for simplifying trigonometric expressions. By using the relationships between the sine, cosine, and tangent functions, you can often rewrite an expression in a more compact form. For example, if you have an expression like $\sin^2(x) + \cos^2(x)$, you can use the identity $\sin^2(x) + \cos^2(x) = 1$ to simplify it to 1. Similarly, you can use the other Pythagorean identities to manipulate and simplify more complex trigonometric expressions involving multiple functions.
• Analyze how Pythagorean identities can be used to prove other trigonometric identities.
  • Pythagorean identities form the foundation for proving many other trigonometric identities. By starting with the fundamental Pythagorean identities and applying algebraic manipulations, you can derive more complex identities. For instance, using the identity $\sin^2(x) + \cos^2(x) = 1$, you can prove the identity $\sec^2(x) = 1 + \tan^2(x)$. Similarly, the Pythagorean identities can be used as a basis to prove trigonometric identities involving the reciprocal functions (cosecant, secant, and cotangent) as well as compound angle identities and double-angle identities. The versatility of Pythagorean identities makes them essential for establishing and verifying a wide range of trigonometric relationships.

Related terms

Pythagorean Theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Trigonometric Functions:

The trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) are functions that describe the relationships between the angles and side lengths of a right triangle.

Unit Circle:

The unit circle is a circle with a radius of 1 unit, centered at the origin of the coordinate plane. It is used to define and visualize the trigonometric functions.