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 3.3.58b
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.3.58bCapitolo 3, Problema 3.3.58b
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
Guida verificata passo dopo passo1
Identify the given information: \( \sec \alpha = -3 \) and the interval \( \frac{\pi}{2} < \alpha < \pi \). Recall that \( \sec \alpha = \frac{1}{\cos \alpha} \).
Find \( \cos \alpha \) by taking the reciprocal of \( \sec \alpha \): \( \cos \alpha = \frac{1}{\sec \alpha} = \frac{1}{-3} = -\frac{1}{3} \).
Since \( \alpha \) is in the interval \( \frac{\pi}{2} < \alpha < \pi \), which corresponds to the second quadrant, note that cosine values are negative there, confirming the sign of \( \cos \alpha \).
Use the double-angle formula for cosine to find \( \cos 2\alpha \): \[ \cos 2\alpha = 2 \cos^2 \alpha - 1 \].
Substitute \( \cos \alpha = -\frac{1}{3} \) into the double-angle formula and simplify the expression to find the exact value of \( \cos 2\alpha \).
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Secant and Cosine Relationship
Secant (sec) is the reciprocal of cosine (cos), so sec α = 1/cos α. Given sec α = -3, we find cos α by taking the reciprocal, resulting in cos α = -1/3. Understanding this reciprocal relationship is essential to convert between secant and cosine values.
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Percorso guidato
6:22
Graphs of Secant and Cosecant Functions
Angle Interval and Sign of Trigonometric Functions
The interval π/2 < α < π places α in the second quadrant, where cosine values are negative. This information confirms the sign of cos α, ensuring the correct value is chosen when solving for cosine or related functions.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Half-Angle Formulas
To find cos(α/2), use the half-angle identity: cos(α/2) = ±√[(1 + cos α)/2]. The sign depends on the quadrant of α/2. Applying this formula allows calculation of the exact value of cos(α/2) from the known cos α.
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Percorso guidato
6:36Quadratic Formula
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:
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
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