Lesson 21 Simplifying and Solving with Pythagorean Identities

Front

What is the Pythagorean Identity?

Back

The Pythagorean Identity states that for any angle \( \theta \), \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

Front

How can you derive the Pythagorean Identities from the unit circle?

Back

The Pythagorean Identities can be derived from the definition of sine and cosine on the unit circle, where the radius is 1. For any angle \( \theta \), the coordinates are (\( \cos(\theta) \), \( \sin(\theta) \)), leading to the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

Front

What is the value of \( \sin(0) \)?

Back

The value of \( \sin(0) \) is 0.

Front

What is the value of \( \cos(0) \)?

Back

The value of \( \cos(0) \) is 1.

Front

What is the value of \( \tan(0) \)?

Back

The value of \( \tan(0) \) is 0.

Front

What is the relationship between sine and cosine?

Back

Sine and cosine are co-functions, meaning \( \sin(\theta) = \cos(90^\circ - \theta) \) or \( \sin(\theta) = \cos(\frac{\pi}{2} - \theta) \).

Front

What is the double angle formula for sine?

Back

The double angle formula for sine is \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \).

Front

What is the double angle formula for cosine?

Back

The double angle formula for cosine is \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \).

Front

How do you solve \( \sin(x) = \frac{1}{2} \) for \( x \) in the interval \( [0, 2\pi) \)?

Back

The solutions are \( x = \frac{\pi}{6} \) and \( x = \frac{5\pi}{6} \).

Front

How do you solve \( \cos(x) = 0 \) for \( x \) in the interval \( [0, 2\pi) \)?

Back

The solutions are \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).