Domanda del libro di testo
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.
891
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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 3.2.57b
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.2.57bCapitolo 3, Problema 3.2.57b
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.
Guida verificata passo dopo passo1
Identify the given information: \(\sin \alpha = \frac{3}{5}\) with \(\alpha\) in quadrant I, and \(\sin \beta = \frac{5}{13}\) with \(\beta\) in quadrant II.
Recall the formula for the sine of a sum: \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\).
Find \(\cos \alpha\) using the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\). Since \(\alpha\) is in quadrant I, \(\cos \alpha\) is positive. Calculate \(\cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(\frac{3}{5}\right)^2}\).
Find \(\cos \beta\) similarly using \(\cos \beta = \pm \sqrt{1 - \sin^2 \beta} = \pm \sqrt{1 - \left(\frac{5}{13}\right)^2}\). Since \(\beta\) is in quadrant II, \(\cos \beta\) is negative.
Substitute the values of \(\sin \alpha\), \(\cos \alpha\), \(\sin \beta\), and \(\cos \beta\) into the formula \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\) to find the exact value.
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Sine Addition Formula
The sine addition formula states that sin(α + β) = sin α cos β + cos α sin β. This identity allows us to find the sine of the sum of two angles using the sines and cosines of the individual angles, which is essential for solving the problem.
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Quadratic Formula
Determining Cosine from Sine and Quadrant
Given sin α and sin β along with their quadrants, we use the Pythagorean identity cos²θ = 1 - sin²θ to find cos α and cos β. The sign of cosine depends on the quadrant: positive in quadrant I and negative in quadrant II, which affects the final calculation.
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Quadrant Sign Rules for Trigonometric Functions
The signs of sine and cosine functions vary by quadrant: in quadrant I, both sine and cosine are positive; in quadrant II, sine is positive and cosine is negative. Correctly applying these sign rules is crucial to accurately compute sin(α + β).
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6:04Introduction to Trigonometric Functions
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In Exercises 55–58, use the given information to find the exact value of each of the following:
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Domanda del libro di testo
In Exercises 57–64, find the exact value of the following under the given conditions:
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In Exercises 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
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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°
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Domanda del libro di testo
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.
848
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