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

검증된 단계별 안내

Recall the Pythagorean identity in trigonometry, which states that for any angle \( \theta \), the following equation holds: \[ \sin^{2} \theta + \cos^{2} \theta = 1 \]

Understand that \( \sin^{2} \theta \) means \( (\sin \theta)^{2} \) and similarly for \( \cos^{2} \theta \). Both sine and cosine values range between -1 and 1, so their squares range between 0 and 1.

Since both \( \sin^{2} \theta \) and \( \cos^{2} \theta \) are non-negative and their sum is always exactly 1, check if the given statement \( \sin^{2} \theta + \cos^{2} \theta = 2 \) can ever be true.

Consider the maximum possible values of \( \sin^{2} \theta \) and \( \cos^{2} \theta \). The maximum value for each is 1, but since they are complementary in the identity, their sum cannot exceed 1.

Conclude that the statement \( \sin^{2} \theta + \cos^{2} \theta = 2 \) is impossible because it contradicts the fundamental Pythagorean identity.

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주요 개념

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Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²θ + cos²θ = 1. This fundamental trigonometric identity is derived from the Pythagorean theorem and holds true for all real values of θ.

Range of Sine and Cosine Functions

The sine and cosine functions each have values ranging between -1 and 1. Consequently, their squares, sin²θ and cos²θ, range from 0 to 1, which restricts the possible sums of these squares.

Evaluating the Possibility of a Statement

To determine if a trigonometric statement is possible, compare it against known identities and function ranges. Since sin²θ + cos²θ always equals 1, the statement sin²θ + cos²θ = 2 is impossible.