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 5.54

Capitolo 6, Problema 5.54

Verify that each equation is an identity.
(sin² θ)/cos θ = sec θ - cos θ

Guida verificata passo dopo passo

Start by rewriting the right-hand side of the equation using trigonometric identities: \( \sec \theta = \frac{1}{\cos \theta} \). So, the right-hand side becomes \( \frac{1}{\cos \theta} - \cos \theta \).

Combine the terms on the right-hand side over a common denominator: \( \frac{1}{\cos \theta} - \cos \theta = \frac{1 - \cos^2 \theta}{\cos \theta} \).

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Therefore, \( 1 - \cos^2 \theta = \sin^2 \theta \).

Substitute \( \sin^2 \theta \) for \( 1 - \cos^2 \theta \) in the expression: \( \frac{\sin^2 \theta}{\cos \theta} \).

Observe that the left-hand side of the original equation is \( \frac{\sin^2 \theta}{\cos \theta} \), which matches the transformed right-hand side, thus verifying the identity.

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Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides of the equation are defined. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for verifying equations and simplifying trigonometric expressions.

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Fundamental Trigonometric Identities

Reciprocal Functions

Reciprocal functions in trigonometry relate the sine, cosine, and tangent functions to their reciprocals: cosecant (csc), secant (sec), and cotangent (cot). For example, sec θ is defined as 1/cos θ. Recognizing these relationships is essential for manipulating and verifying trigonometric equations.

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Secant, Cosecant, & Cotangent on the Unit Circle

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations using algebraic rules. This includes factoring, combining like terms, and applying identities to transform one side of an equation to match the other. Mastery of these techniques is necessary for verifying trigonometric identities effectively.

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Algebraic Operations on Vectors

Pratica correlata

Domanda del libro di testo

Verify that each equation is an identity.

(sec α + csc α) (cos α - sin α) = cot α - tan α

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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.

-tan x cos 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

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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.

(sin θ - cos θ) (csc θ + sec θ)

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Domanda del libro di testo

Perform each transformation. See Example 2.

Write cot x in terms of csc x.

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Domanda del libro di testo

Determine whether the positive or negative square root should be selected.

cos 58° = ±√ (1 + cos 116°)/2]

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