Domanda del libro di testo
Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan 3x = (√3)/3
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 3, Problema 3
In Exercises 1–60, verify each identity. tan (-x) cos x = -sin x
Guida verificata passo dopo passo1
Start by recalling the trigonometric identity for the tangent of a negative angle: \( \tan(-x) = -\tan(x) \).
Substitute \( \tan(-x) \) with \( -\tan(x) \) in the given expression: \( \tan(-x) \cos(x) = -\tan(x) \cos(x) \).
Use the definition of tangent in terms of sine and cosine: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
Substitute \( \tan(x) \) with \( \frac{\sin(x)}{\cos(x)} \) in the expression: \( -\tan(x) \cos(x) = -\left(\frac{\sin(x)}{\cos(x)}\right) \cos(x) \).
Simplify the expression by canceling \( \cos(x) \) in the numerator and denominator: \( -\sin(x) \). This shows that \( \tan(-x) \cos(x) = -\sin(x) \), verifying the identity.
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Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for simplifying expressions and verifying equations in trigonometry.
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Percorso guidato
5:32
Fundamental Trigonometric Identities
Tangent Function and Its Properties
The tangent function, defined as the ratio of the sine and cosine functions (tan(x) = sin(x)/cos(x)), has specific properties, including periodicity and symmetry. Notably, tan(-x) = -tan(x), which reflects the odd nature of the tangent function. This property is essential for manipulating and verifying trigonometric identities.
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5:43Introduction to Tangent Graph
Negative Angle Identities
Negative angle identities express the values of trigonometric functions for negative angles. For example, sin(-x) = -sin(x) and cos(-x) = cos(x). These identities help in transforming expressions involving negative angles into more manageable forms, which is particularly useful in verifying identities like the one presented in the question.
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Percorso guidato
05:06Double Angle Identities
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Domanda del libro di testo
Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). sec(3θ/2) = - 2
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Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 2 cos² x + sin x - 1 = 0
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Domanda del libro di testo
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. tan(θ/2)
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Domanda del libro di testo
Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan(x/2) = √3
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Domanda del libro di testo
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. cos(α/2)
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