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 5.1.64

Capitolo 6, Problema 5.1.64

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.
1 + cot(-θ)/cot(-θ)

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Recall the definition of cotangent in terms of sine and cosine: \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\). Therefore, \(\cot(-\theta) = \frac{\cos(-\theta)}{\sin(-\theta)}\).

Use the even-odd properties of cosine and sine: \(\cos(-\theta) = \cos(\theta)\) (cosine is even) and \(\sin(-\theta) = -\sin(\theta)\) (sine is odd). Substitute these into the expression for \(\cot(-\theta)\) to get \(\cot(-\theta) = \frac{\cos(\theta)}{-\sin(\theta)} = -\frac{\cos(\theta)}{\sin(\theta)}\).

Rewrite the original expression \(1 + \frac{\cot(-\theta)}{\cot(-\theta)}\) by substituting the simplified form of \(\cot(-\theta)\): \(1 + \frac{-\frac{\cos(\theta)}{\sin(\theta)}}{-\frac{\cos(\theta)}{\sin(\theta)}}\).

Simplify the fraction \(\frac{-\frac{\cos(\theta)}{\sin(\theta)}}{-\frac{\cos(\theta)}{\sin(\theta)}}\). Since numerator and denominator are the same, this fraction simplifies to 1.

Combine the terms: \(1 + 1 = 2\). The expression is now simplified with no quotients and only sine and cosine functions of \(\theta\).

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Trigonometric Functions and Their Definitions

Understanding the basic trigonometric functions—sine, cosine, and cotangent—is essential. Cotangent is defined as the ratio of cosine to sine (cot θ = cos θ / sin θ). Expressing all functions in terms of sine and cosine helps simplify complex expressions.

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Introduction to Trigonometric Functions

Even-Odd Properties of Trigonometric Functions

Knowing how trigonometric functions behave with negative angles is crucial. Sine is an odd function (sin(-θ) = -sin θ), while cosine is even (cos(-θ) = cos θ). Cotangent, being a quotient of cosine and sine, is an odd function (cot(-θ) = -cot θ). This helps simplify expressions involving negative angles.

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Even and Odd Identities

Simplification of Trigonometric Expressions

Simplifying expressions involves rewriting quotients as products or sums without fractions, and combining like terms. The goal is to express the entire expression using only sine and cosine functions of θ, eliminating quotients and negative angles for clarity and ease of further manipulation.

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Simplifying Trig Expressions

Pratica correlata

Domanda del libro di testo

Verify that each equation is an identity.

(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t

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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.

csc θ - sin θ

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Domanda del libro di testo

Verify that each equation is an identity.

(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1

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Domanda del libro di testo

Find the remaining five trigonometric functions of θ.

cos θ = -1/4, sin θ > 0

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