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
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.2.61c
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.2.61cCapitolo 3, Problema 3.2.61c
In Exercises 57–64, find the exact value of the following under the given conditions:
c. tan (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 the signs and values of \(\sin \alpha\) and \(\cos \beta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), considering the quadrant of each angle.
Calculate \(\sin \alpha\) by using \(\sin \alpha = -\sqrt{1 - \cos^2 \alpha}\) since \(\alpha\) is in quadrant IV where sine is negative.
Calculate \(\cos \beta\) by using \(\cos \beta = -\sqrt{1 - \sin^2 \beta}\) since \(\beta\) is in quadrant III where cosine is negative.
Use the angle addition formula for tangent: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\), where \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\) and \(\tan \beta = \frac{\sin \beta}{\cos \beta}\). Substitute the values found to express \(\tan(\alpha + \beta)\).
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Trigonometric Ratios and Quadrants
Trigonometric ratios (sine, cosine, tangent) relate the angles of a triangle to the ratios of its sides. The sign of these ratios depends on the quadrant in which the angle lies. For example, in quadrant IV, cosine is positive and sine is negative, while in quadrant III, both sine and cosine are negative.
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Percorso guidato
6:36
Quadratic Formula
Sum of Angles Formula for Tangent
The tangent of the sum of two angles α and β is given by tan(α + β) = (tan α + tan β) / (1 - tan α tan β). This formula allows us to find the exact value of tan(α + β) using the individual tangents of α and β, which can be derived from their sine and cosine values.
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Percorso guidato
4:47Sum and Difference of Tangent
Finding Missing Trigonometric Values Using Pythagorean Identity
Given one trigonometric ratio and the quadrant, the other ratios 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. This step is essential to compute tan α and tan β.
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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:
c. tan (α + β)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.
861
views
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 55–58, use the given information to find the exact value of each of the following:
c. tan(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
Domanda del libro di testo
In Exercises 55–58, use the given information to find the exact value of each of the following:
c. tan(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅
632
views