Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.50

Capitolo 6, Problema 5.50

Verify that each equation is an identity.
sin² β (1 + cot² β) = 1

Guida verificata passo dopo passo

Start by recalling the Pythagorean identity: \(1 + \cot^2 \beta = \csc^2 \beta\).

Substitute \(1 + \cot^2 \beta\) with \(\csc^2 \beta\) in the given equation: \(\sin^2 \beta \cdot \csc^2 \beta = 1\).

Remember that \(\csc \beta = \frac{1}{\sin \beta}\), so \(\csc^2 \beta = \frac{1}{\sin^2 \beta}\).

Substitute \(\csc^2 \beta\) with \(\frac{1}{\sin^2 \beta}\) in the equation: \(\sin^2 \beta \cdot \frac{1}{\sin^2 \beta} = 1\).

Simplify the expression: \(\sin^2 \beta \cdot \frac{1}{\sin^2 \beta} = 1\), which confirms the identity.

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Pythagorean Identity

The Pythagorean identity states that for any angle β, sin² β + cos² β = 1. This fundamental relationship between sine and cosine is crucial for verifying trigonometric identities, as it allows us to express one function in terms of another, simplifying the equation.

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Percorso guidato

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Pythagorean Identities

Cotangent Function

The cotangent function, defined as cot β = cos β / sin β, is the reciprocal of the tangent function. The identity cot² β = cos² β / sin² β is essential for manipulating equations involving cotangent, particularly when combined with the Pythagorean identity to simplify expressions.

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Introduction to Cotangent Graph

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include reciprocal, quotient, and Pythagorean identities. Understanding these identities is key to verifying equations, as they provide the necessary tools to transform and simplify expressions.

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Percorso guidato

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Fundamental Trigonometric Identities

Pratica correlata

Domanda del libro di testo

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

tan(-θ)/sec θ

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Domanda del libro di testo

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

sec x/csc x

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

570

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Domanda del libro di testo

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sec θ - 1) (sec θ + 1)

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Domanda del libro di testo

Verify that each equation is an identity.

(tan² α + 1)/ sec α = sec α

797

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Domanda del libro di testo

Verify that each equation is an identity.

(sec α - tan α)² = (1 - sin α)/(1 + sin α)

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