Pythagorean Identities Lesson

Pythagorean Identities are a set of trigonometric identities that are derived from the Pythagorean Theorem. These identities are fundamental in trigonometry and are used extensively in other mathematical contexts as well. The Pythagorean Identities are derived from the Pythagorean Theorem, which states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs). This theorem can be applied to trigonometric ratios for a right triangle, which results in Pythagorean identities. There are three main Pythagorean Identities, which involve sine, cosine and tangent functions:

  1. sin2 Θ + cos2 Θ = 1

a. The most significant Pythagorean Identity, which gives the relation between sine and cosine

  1. sec2 Θ – tan2 Θ = 1

a. Gives the relation between secant and tangent

  1. csc2 Θ – cot2 Θ = 1

a. Gives the relation between cosecant and cotangent

Each Pythagorean Identity can be written in two other forms using algebraic operations.

  1. sin2 Θ + cos2 Θ = 1

a. sin2 Θ = 1 – cos2 Θ

b. cos2 Θ = 1 – sin2 Θ

  1. sec2 Θ – tan2 Θ = 1

a. sec2 Θ = 1 + tan2 Θ

b. tan2 Θ = sec2 Θ – 1

  1. csc2 Θ – cot2 Θ = 1

a. csc2 Θ = 1 + cot2 Θ

b. cot2 Θ = csc2 Θ – 1

These identities and their various forms are essential tools in trigonometry and calculus, they allow simplification of trigonometric expressions, solving trigonometric equations, and proving various mathematical theorems.

Most often Pythagorean Identities are used to prove other trigonometric identities.

Examples

Example 1: Prove the identity (sec2 Θ – 1) cos2 Θ = sin2 Θ

Solution: Start by recognizing that (sin2 Θ – 1) is a Pythagorean Identity, and can be rewritten as tan2 Θ. Then we can rewrite

tan2 Θ = sin2 Θ / cos2 Θ

Now we can cancel the cos2 Θ from the numerator and denominator when we multiply the left side, and we are left with sin2 Θ on the left, which equals the right hand side.

sec working through problem.

Example 2: Prove the identity sin Θ – sin Θ cos2 Θ = cos3 Θ

Solution: To prove this identity is true, we will begin by factoring out a sin Θ on the left side of the equation. Then we will have sin Θ (1 – cos2 Θ), in which case we notice that we have a Pythagorean Identity, 1 – cos2 Θ, which we can rewrite as sin2 Θ.Lastly, that would leave us with sin Θ (sin2 Θ) which equals sin3 Θ.

Problem Work through

Example 3: Prove (csc Θ – cot Θ)(csc Θ + cot Θ) = 1

Solution: Start by using the distributive property to multiply the two groups together, and combine like terms. This will result in csc2 Θ – cot2 Θ = 1. Now I will use one of my Pythagorean Identities, csc2 Θ = 1 + cot2 Θ to rewrite the equation as 1 + cot2 Θ – cot2 Θ = 1. In which, the positive and negative cot2 Θ will cancel, and you are left with 1 = 1.

Fully Problem Explanation

FAQs on Pythagorean Identities

  1. What are Pythagorean identities?

Pythagorean identities are a set of fundamental trigonometric identities derived from the Pythagorean theorem, which relates the sides of a right triangle. These identities establish relationships between the sine, cosine, and tangent functions.

  1. How many Pythagorean identities are there?

There are three main Pythagorean identities:

A. sin2(x) + cos2(x) = 1
B. 1 + tan2(x) = sec2(x)
C. cot2(x) + 1 = csc2(x)

  1. What is the significance of Pythagorean identities?

Pythagorean identities are fundamental tools in trigonometry. They allow for the simplification of trigonometric expressions, solving trigonometric equations, and proving various mathematical theorems. They also establish deep connections between trigonometric functions and geometric concepts, particularly the geometry of right triangles.

  1. How are Pythagorean identities derived?

Pythagorean identities can be derived from the Pythagorean theorem, which 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. By considering the definitions of trigonometric functions in terms of the sides of a right triangle, these identities can be derived algebraically.

  1. What are some applications of Pythagorean identities?

Pythagorean identities are used extensively in various branches of mathematics, physics, engineering, and other scientific fields. They are particularly useful in calculus, where they are employed in integration, differentiation, and solving differential equations involving trigonometric functions. They also find applications in signal processing, wave analysis, and geometric modeling, among other areas.

  1. Are there variations or extensions of Pythagorean identities?

While the three main Pythagorean identities are the most commonly used, there are additional identities that can be derived from them or from other trigonometric relationships. These may include reciprocal identities, quotient identities, and Pythagorean identities involving hyperbolic trigonometric functions.