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Chapter 1: Problem 5

What are the three Pythagorean identities for the trigonometric functions?

Short Answer

Expert verified
A: The three Pythagorean identities for trigonometric functions are: 1. Sine and Cosine relation: $\sin^2(\theta) + \cos^2(\theta) = 1$ 2. Tangent and Secant relation: $\tan^2(\theta) + 1 = \sec^2(\theta)$ 3. Cotangent and Cosecant relation: $\cot^2(\theta) + 1 = \csc^2(\theta)$

Step by step solution

01

Pythagorean Identity 1: Sine and Cosine relation

The first identity relates the square of the sine and cosine functions of an angle. We can derive the identity using the Pythagorean theorem and the definition of sine and cosine. The formula is: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
02

Pythagorean Identity 2: Tangent and Secant relation

The second identity connects the tangent and secant functions of an angle. We can derive this identity using the first identity and dividing both sides by the square of the cosine function. The formula is: \[ \tan^2(\theta) + 1 = \sec^2(\theta) \]
03

Pythagorean Identity 3: Cotangent and Cosecant relation

The third identity links the cotangent and cosecant functions of an angle. We can derive this identity using the first identity and dividing both sides by the square of the sine function. The formula is: \[ \cot^2(\theta) + 1 = \csc^2(\theta) \] These are the three Pythagorean identities for the trigonometric functions.

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