Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin² x - 2 cos x - 2 = 0
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 3, Problema 64c
In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .
Guida verificata passo dopo passo1
Identify the given information: \( \sin \alpha = \frac{5\pi}{6} \) with \( \frac{3\pi}{2} < \alpha < \pi \), and \( \tan \beta = \frac{7}{2} \) with \( \pi < \beta < \frac{3\pi}{2} \). Note that the interval for \( \alpha \) seems inconsistent since \( \frac{3\pi}{2} > \pi \). Verify the correct interval for \( \alpha \) before proceeding.
Recall the formula for the tangent of a sum:
\(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\)
Find \( \tan \alpha \) using the given \( \sin \alpha \) and the Pythagorean identity. Since \( \sin \alpha = \frac{5\pi}{6} \), use:
\(\cos \alpha = \pm \sqrt{1 - \sin^2 \alpha}\)
Determine the sign of \( \cos \alpha \) based on the quadrant of \( \alpha \). Then calculate \( \(\tan\) \(\alpha\) = \(\frac{\sin \alpha}{\cos \alpha}\)$.
Use the given \( \tan \beta = \frac{7}{2} \) directly, noting the sign of \( \tan \beta \) based on the quadrant of \( \beta \).
Substitute \( \tan \alpha \) and \( \tan \beta \) into the tangent sum formula:
\(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\)
Simplify the expression to find the exact value.
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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 is essential for solving the problem.
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Percorso guidato
4:47
Sum and Difference of Tangent
Determining the Sign and Value of Trigonometric Functions in Specific Quadrants
Knowing the quadrant in which an angle lies helps determine the sign (positive or negative) of sine, cosine, and tangent values. Since α and β are restricted to certain intervals, this information is crucial to correctly evaluate the trigonometric functions and avoid sign errors.
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5:30Trig Values in Quadrants II, III, & IV
Finding Tangent from Sine and Quadrant Information
Given sin α and the quadrant of α, we can find cos α using the Pythagorean identity cos²α = 1 - sin²α, then compute tan α = sin α / cos α. This step is necessary when only sine is provided, enabling the use of the sum formula for tangent.
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Percorso guidato
5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
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In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 5 cos² x - 3 = 0
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In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .
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Domanda del libro di testo
Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). cos 4x = ﹣√3 / 2
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