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.RE.10

Capitolo 6, Problema 5.RE.10

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc θ - sin θ

Guida verificata passo dopo passo

Recall that the cosecant function is the reciprocal of the sine function, so we can write \(\csc \theta\) as \(\frac{1}{\sin \theta}\).

Rewrite the expression \(\csc \theta - \sin \theta\) as \(\frac{1}{\sin \theta} - \sin \theta\) to express everything in terms of \(\sin \theta\).

To combine the terms into a single expression without quotients, find a common denominator, which is \(\sin \theta\), and rewrite the expression as \(\frac{1}{\sin \theta} - \frac{\sin^2 \theta}{\sin \theta}\).

Combine the fractions over the common denominator: \(\frac{1 - \sin^2 \theta}{\sin \theta}\).

Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to replace \(1 - \sin^2 \theta\) with \(\cos^2 \theta\), resulting in \(\frac{\cos^2 \theta}{\sin \theta}\). This expression is now in terms of \(\sin \theta\) and \(\cos \theta\) only, with no quotients involving other trigonometric functions.

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.

Reciprocal Identities

Reciprocal identities relate trigonometric functions to each other by expressing one as the reciprocal of another. For example, csc θ is the reciprocal of sin θ, meaning csc θ = 1/sin θ. Understanding these identities allows rewriting expressions to involve only sine and cosine.

Video consigliato:

Percorso guidato

6:25

Pythagorean Identities

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting them to eliminate complex fractions or quotients and combining like terms. This process often uses algebraic manipulation and identities to express the function in a simpler or more standard form, such as removing quotients and using only sin θ and cos θ.

Video consigliato:

Percorso guidato

6:36

Simplifying Trig Expressions

Expressing Functions in Terms of sin θ and cos θ

Many trigonometric functions can be expressed using only sine and cosine, which are the fundamental trigonometric functions. For example, csc θ = 1/sin θ, sec θ = 1/cos θ, and tan θ = sin θ/cos θ. Writing expressions solely in terms of sin θ and cos θ helps in standardizing and simplifying trigonometric problems.

Video consigliato:

Percorso guidato

3:43

Finding the Domain of an Equation

Pratica correlata

Domanda del libro di testo

Find values of the sine and cosine functions for each angle measure.

2y, given sec y = -5/3, sin y > 0

689

views

Domanda del libro di testo

Use the given information to cos(x - y).

cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

821

views

Domanda del libro di testo

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

cos 75°

726

views

Domanda del libro di testo

For each expression in Column I, choose the expression from Column II that completes an identity.

4. cot 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

699