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Chapter 7: Problem 2

State the three Pythagorean identities.

Short Answer

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Question: State the three Pythagorean identities in trigonometry. Answer: The three Pythagorean identities in trigonometry are: 1. \(\sin^2(\theta) + \cos^2(\theta) = 1\) 2. \(1 + \tan^2(\theta) = \sec^2(\theta)\) 3. \(1 + \cot^2(\theta) = \csc^2(\theta)\)

Step by step solution

01

Pythagorean Identity 1

The first identity is derived from the basic definition of sine and cosine in relation to a right triangle. By applying the Pythagorean theorem (a^2 + b^2 = c^2), we arrive at the following equation: \[\sin^2(\theta) + \cos^2(\theta) = 1\]
02

Pythagorean Identity 2

The second identity is derived by dividing each term in the first identity by \(\cos^2(\theta)\). This effectively changes the sine and cosine functions into tangent and secant functions respectively. The second identity is: \[1 + \tan^2(\theta) = \sec^2(\theta)\]
03

Pythagorean Identity 3

The third identity is derived by dividing each term in the first identity by \(\sin^2(\theta)\). This effectively changes the sine and cosine functions into cotangent and cosecant functions respectively. The third identity is: \[1 + \cot^2(\theta) = \csc^2(\theta)\] These are the three Pythagorean identities that relate the various trigonometric functions.

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Most popular questions from this chapter

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