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
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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 3.3.55a
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.3.55aCapitolo 3, Problema 3.3.55a
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°
Guida verificata passo dopo passo1
Identify the quadrant where the angle \( \alpha \) lies. Since \( 180^\circ < \alpha < 270^\circ \), \( \alpha \) is in the third quadrant, where both sine and cosine values are negative, but tangent is positive.
Recall the given value: \( \tan \alpha = \frac{4}{3} \). Use the identity \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) to relate sine and cosine.
Set \( \sin \alpha = 4k \) and \( \cos \alpha = 3k \) for some constant \( k \), because the ratio of sine to cosine must be \( \frac{4}{3} \). Since \( \alpha \) is in the third quadrant, both sine and cosine are negative, so \( \sin \alpha = -4k \) and \( \cos \alpha = -3k \).
Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to solve for \( k \):
\[
(-4k)^2 + (-3k)^2 = 1 \\
16k^2 + 9k^2 = 1 \\
25k^2 = 1 \\
k^2 = \frac{1}{25} \\
k = \frac{1}{5}
\]
Substitute \( k = \frac{1}{5} \) back into \( \sin \alpha = -4k \) to find \( \sin \alpha \):
\[
\sin \alpha = -4 \times \frac{1}{5} = -\frac{4}{5}
\]
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Trigonometric Ratios and Their Signs in Quadrants
Trigonometric functions like sine and tangent have specific signs depending on the quadrant of the angle. Since 180° < α < 270°, α lies in the third quadrant where sine is negative and tangent is positive. Understanding these sign rules is essential to determine the correct value of sin(α/2).
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Percorso guidato
6:36
Quadratic Formula
Half-Angle Formulas
Half-angle formulas allow calculation of trigonometric values for half of a given angle. For sine, the formula is sin(α/2) = ±√((1 - cos α)/2). Choosing the correct sign depends on the quadrant of α/2, which must be determined from the original angle α.
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Percorso guidato
6:36Quadratic Formula
Relationship Between Tangent and Cosine
Given tan α, you can find cos α using the identity tan α = sin α / cos α and the Pythagorean identity sin² α + cos² α = 1. This step is crucial because the half-angle formula for sine requires the value of cos α.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Pratica correlata
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
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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 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
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)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.
950
views