Domanda del libro di testo
In Exercises 55–58, use the given information to find the exact value of each of the following:
b. cos(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅
607
views
Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
Tutti i libri di testo
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.2.61b
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.2.61bCapitolo 3, Problema 3.2.61b
In Exercises 57–64, find the exact value of the following under the given conditions:
b. sin (α + β)
cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.
Guida verificata passo dopo passo1
Identify the given information: \(\cos \alpha = \frac{8}{17}\) with \(\alpha\) in quadrant IV, and \(\sin \beta = -\frac{1}{2}\) with \(\beta\) in quadrant III.
Determine \(\sin \alpha\) using the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\). Since \(\cos \alpha = \frac{8}{17}\), calculate \(\sin \alpha = \pm \sqrt{1 - \left(\frac{8}{17}\right)^2}\). Because \(\alpha\) is in quadrant IV, where sine is negative, choose the negative root.
Determine \(\cos \beta\) using the Pythagorean identity \(\sin^2 \beta + \cos^2 \beta = 1\). Since \(\sin \beta = -\frac{1}{2}\), calculate \(\cos \beta = \pm \sqrt{1 - \left(-\frac{1}{2}\right)^2}\). Because \(\beta\) is in quadrant III, where cosine is negative, choose the negative root.
Use the angle sum identity for sine: \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\).
Substitute the values of \(\sin \alpha\), \(\cos \beta\), \(\cos \alpha\), and \(\sin \beta\) into the identity and simplify to find the exact value of \(\sin(\alpha + \beta)\).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
8mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Trigonometric Angle Sum Identity
The sine of a sum of two angles, sin(α + β), can be found using the identity sin(α + β) = sin α cos β + cos α sin β. This formula allows the expression of the sine of a combined angle in terms of the sines and cosines of the individual angles.
Video consigliato:
Percorso guidato
2:25
Verifying Identities with Sum and Difference Formulas
Determining Signs of Trigonometric Functions by Quadrant
The sign of sine and cosine values depends on the quadrant in which the angle lies. In quadrant IV, cosine is positive and sine is negative; in quadrant III, both sine and cosine are negative. This knowledge is essential to correctly assign signs when calculating unknown trigonometric values.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
Using Pythagorean Identity to Find Missing Values
When one trigonometric value is given, the other can be found using the Pythagorean identity sin²θ + cos²θ = 1. For example, if cos α is known, sin α can be found by sin α = ±√(1 - cos²α), with the sign determined by the quadrant of α.
Video consigliato:
Percorso guidato
6:25Pythagorean Identities
Pratica correlata
Domanda del libro di testo
In Exercises 57–64, find the exact value of the following under the given conditions:
b. sin (α + β)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.
886
views
Domanda del libro di testo
In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
tan α = ﹣3/4, α lies in quadrant II, and cos β = 1/3, β lies in quadrant I.
965
views
Domanda del libro di testo
In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅
989
views
Domanda del libro di testo
In Exercises 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
tan α = 4/3, 180° < α < 270°
650
views
Domanda del libro di testo
In Exercises 55–58, use the given information to find the exact value of each of the following:
b. cos(α/2)
tan α = 4/3, 180° < α < 270°
666
views