CONCEPT PREVIEW  Determine whether each statement is possible or impossible. sin² θ + cos² θ = 2

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, the sum of the squares of the sine and cosine functions equals one: sin² θ + cos² θ = 1. This fundamental identity is crucial in trigonometry and serves as a basis for many other trigonometric equations and transformations.

Range of Sine and Cosine Functions

The sine and cosine functions have a range of values between -1 and 1. This means that the maximum value of sin² θ and cos² θ is 1, making the maximum possible value of sin² θ + cos² θ equal to 2 only if both functions equal 1 simultaneously, which is impossible for any angle θ.

Trigonometric Values at Specific Angles

Trigonometric functions take specific values at certain angles. For example, sin(90°) = 1 and cos(90°) = 0. However, there is no angle θ for which both sin² θ and cos² θ can equal 1 at the same time, reinforcing that the equation sin² θ + cos² θ = 2 is impossible.