Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 64b

Chapter 3, Problem 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 .

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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