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Pythagorean Identities

Pythagorean Identities

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Understanding Pythagorean Identities: The Key to Mastering Trigonometry

Pythagorean identities are a set of equations that are vital in the study of trigonometry. These equations are derived from Pythagoras' theorem and play a crucial role in solving various mathematical problems. In this article, we will explore the three fundamental Pythagorean identities and their derivation process.

To understand Pythagorean identities, it is essential to first familiarize ourselves with Pythagoras' theorem. This theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This forms the basis for the first Pythagorean identity.

The First Pythagorean Identity: This identity is expressed as sin^2(𝛉) +cos^2(𝛉) =1 and can be derived using Pythagoras' theorem and the concept of the unit circle. By substituting the values of the sides of a right-angled triangle into the equation and simplifying, we obtain the given form.

The Second Pythagorean Identity: To obtain the second Pythagorean identity, we divide the first identity by cos^2(𝛉). This results in the equation tan^2(𝛉)+1 = sec^2(𝛉), where tan𝛉 = sin𝛉/cos𝛉 and sec𝛉 = 1/cos𝛉.

The Third Pythagorean Identity: Similarly, the third Pythagorean identity is derived by dividing the first identity by sin^2(𝛉), giving us the equation 1+cot^2(𝛉) = csc^2(𝛉), where cot𝛉 = cos𝛉/sin𝛉 and csc𝛉 = 1/sin𝛉.

Applying Pythagorean Identities: Examples

Let's take a look at a few examples to better understand the application of Pythagorean identities.

  • Simplifying and Finding a Value: Simplify and find the value of x (0 < x < 2πœ‹). We can use the first Pythagorean identity and rearrange it to get sin^2x = 1-cos^2x. By substituting this into the given expression and simplifying, we get sin x = 1, which leads to the solution x = πœ‹ / 2.
  • Finding the Value of tan x: If cos x = 0.78, what is the value of tan x? We use the relationship tan𝛉 = sin𝛉/cos𝛉 and substitute the given value of cos x to get tan x = 1.28.
  • Solving for x: Given the equation cosec 2x = -2 or 1, find the value of x between 0Β° and 180Β°. In this case, we must use the third Pythagorean identity, 1+cot^2(𝛉) = csc^2(𝛉). By rearranging this identity, we get cot^2(𝛉) = -1, which simplifies to tan𝛉 = Β±i. Solving for x gives us the solutions x = 45Β°, 105Β°, 165Β°.

Summary

In summary, Pythagorean identities are crucial in solving trigonometric equations and understanding the relationship between trigonometric functions. These identities are derived from Pythagoras' theorem and the unit circle, and the three fundamental identities (sin^2(𝛉) +cos^2(𝛉) =1, tan^2(𝝷)+1=sec^2(𝝷), and 1+cot^2(𝝷)=csc^2(𝝷)) serve as powerful tools in solving complex mathematical problems. By practicing and mastering the application of these identities, you can become proficient in solving various trigonometry problems.

Frequently Asked Questions

  • How are Pythagorean identities derived? Pythagorean identities are derived from Pythagoras' theorem and the unit circle.
  • What are Pythagorean identities? Pythagorean identities are equations based on Pythagoras' theorem that are used to solve or simplify trigonometric equations.
  • What are the three Pythagorean identities? The three Pythagorean identities are sin^2(𝛉) +cos^2(𝛉) =1, tan^2(𝝷)+1=sec^2(𝝷), and 1+cot^2(𝝷)=csc^2(𝝷).

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