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Ch. 1 - Trigonometric Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 1 - Trigonometric FunctionsProblema 10
Capitolo 2, Problema 10

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

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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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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.
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Pythagorean Identities

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.
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Graph of Sine and Cosine Function

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.
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Evaluate Composite Functions - Values Not on Unit Circle
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