Question

What are the three Pythagorean identities for the trigonometric functions?

Short answer

Answer: The three Pythagorean identities for trigonometric functions are: 1. \(\sin^2{\theta} + \cos^2{\theta} = 1\) 2. \(\tan^2{\theta} + 1 = \sec^2{\theta}\) 3. \(1 + \cot^2{\theta} = \csc^2{\theta}\)

Step-by-step solution

Pythagorean Identity 1

The first Pythagorean identity is derived from the Pythagorean theorem applied to a right triangle with hypotenuse 1 and with the other two sides being the sine and cosine of the angle \(\theta\). We have: \(\sin^2{\theta} + \cos^2{\theta} = 1\)

Pythagorean Identity 2

The second Pythagorean identity is derived by dividing the first identity by \(\cos^2{\theta}\), we get: \(\tan^2{\theta} + 1 = \sec^2{\theta}\). This identity relates tangent and secant functions.

Pythagorean Identity 3

The third Pythagorean identity is derived by dividing the first identity by \(\sin^2{\theta}\), we get: \(1 + \cot^2{\theta} = \csc^2{\theta}\). This identity relates cotangent and cosecant functions. So, the three Pythagorean identities for the trigonometric functions are: 1. \(\sin^2{\theta} + \cos^2{\theta} = 1\) 2. \(\tan^2{\theta} + 1 = \sec^2{\theta}\) 3. \(1 + \cot^2{\theta} = \csc^2{\theta}\)