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 5.60

Capitolo 6, Problema 5.60

Verify that each equation is an identity.
[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ

Guida verificata passo dopo passo

Step 1: Start by simplifying the left-hand side (LHS) of the equation. Expand \((\sec \theta - \tan \theta)^2\) using the identity \((a - b)^2 = a^2 - 2ab + b^2\).

Step 2: Use the Pythagorean identities \(\sec^2 \theta = 1 + \tan^2 \theta\) and \(\csc^2 \theta = 1 + \cot^2 \theta\) to simplify the expression further.

Step 3: Simplify the denominator \(\sec \theta \csc \theta - \tan \theta \csc \theta\) by expressing \(\sec \theta\) and \(\csc \theta\) in terms of sine and cosine: \(\sec \theta = \frac{1}{\cos \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\).

Step 4: Substitute the simplified expressions back into the LHS and simplify the fraction. Look for common terms that can be canceled out.

Step 5: Verify that the simplified LHS equals the right-hand side (RHS), \(2 \tan \theta\), by ensuring both sides are equivalent after simplification.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Trigonometric Identities

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

Video consigliato:

Percorso guidato

5:32

Fundamental Trigonometric Identities

Secant and Tangent Functions

The secant function (sec θ) is the reciprocal of the cosine function, while the tangent function (tan θ) is the ratio of the sine function to the cosine function. These functions are fundamental in trigonometry and often appear in various identities and equations. Recognizing their relationships helps in manipulating and simplifying expressions involving them.

Video consigliato:

Percorso guidato

6:22

Graphs of Secant and Cosecant Functions

Algebraic Manipulation in Trigonometry

Algebraic manipulation involves rearranging and simplifying expressions to prove identities. This includes factoring, expanding, and combining like terms. In trigonometry, it is essential to apply these techniques to transform one side of an equation into the other, thereby verifying the identity effectively.

Video consigliato:

Percorso guidato

04:12

Algebraic Operations on Vectors

Pratica correlata

Domanda del libro di testo

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

sin (-10°) = ± √[(1 - cos (-20°))/2]

905

views

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

views

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)

777

views

Domanda del libro di testo

Verify that each equation is an identity.

(tan² α + 1)/ sec α = sec α

797

views

Domanda del libro di testo

Match each expression in Column I with its value in Column II.

cos² (π/6) - sin² (π/6)

583