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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TOC.tableOfContents
Capitolo 3, Problema 64b
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 .
Guida verificata passo dopo passo1
Identify the given information: \( \sin \alpha = \frac{5\pi}{6} \) with \( \frac{3\pi}{2} < \alpha < \pi \), and \( \tan \beta = \frac{3\pi}{2} \) with \( \frac{7\pi}{2} < \beta < \text{(missing upper bound)} \). Note that the interval for \( \alpha \) seems inconsistent since \( \frac{3\pi}{2} > \pi \). Verify the intervals and values carefully before proceeding.
Recall the formula for \( \sin(\alpha + \beta) \):
\[
\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
\]
Since \( \sin \alpha \) is given, find \( \cos \alpha \) using the Pythagorean identity:
\[
\cos \alpha = \pm \sqrt{1 - \sin^2 \alpha}
\]
Determine the correct sign of \( \cos \alpha \) based on the quadrant where \( \alpha \) lies.
Use the given \( \tan \beta \) to find \( \sin \beta \) and \( \cos \beta \). Recall that:
\[
\tan \beta = \frac{\sin \beta}{\cos \beta}
\]
Use the Pythagorean identity to express \( \sin \beta \) and \( \cos \beta \) in terms of \( \tan \beta \), and determine their signs based on the quadrant of \( \beta \).
Substitute the values of \( \sin \alpha \), \( \cos \alpha \), \( \sin \beta \), and \( \cos \beta \) into the formula for \( \sin(\alpha + \beta) \) and simplify to find the exact value.
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Sum of Angles Formula for Sine
The sum of angles formula states that sin(α + β) = sin α cos β + cos α sin β. This identity allows us to find the sine of a sum of two angles using the sines and cosines of the individual angles, which is essential when given trigonometric values of α and β separately.
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Percorso guidato
2:25
Verifying Identities with Sum and Difference Formulas
Determining the Sign of Trigonometric Functions Based on Quadrants
The signs of sine, cosine, and tangent depend on the quadrant in which the angle lies. Knowing the interval for α and β helps determine whether sine, cosine, or tangent values are positive or negative, which is crucial for correctly evaluating trigonometric expressions.
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6:04Introduction to Trigonometric Functions
Using Given Trigonometric Ratios to Find Missing Values
Given sin α and tan β, we can use Pythagorean identities to find cos α and cos β. For example, cos α = ±√(1 - sin² α), with the sign determined by the quadrant. Similarly, from tan β, we find sin β and cos β using the identity tan β = sin β / cos β, enabling full evaluation of sin(α + β).
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5:13Finding Missing Angles
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