Ch. 4 - Laws of Sines and Cosines; Vectors

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

Blitzer 3rd Edition

Ch. 4 - Laws of Sines and Cosines; Vectors

Problem 5

Chapter 4, Problem 5

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle.cos 50° cos 20° + sin 50° sin 20°

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Recognize the trigonometric identity for the cosine of a difference: \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \).

Identify the given expression: \( \cos 50^\circ \cos 20^\circ + \sin 50^\circ \sin 20^\circ \).

Compare the given expression with the identity to determine \( \alpha \) and \( \beta \).

Notice that \( \alpha = 50^\circ \) and \( \beta = 20^\circ \).

Write the expression as \( \cos(50^\circ - 20^\circ) \).

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Key Concepts

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

Cosine of Angle Difference Formula

The cosine of the difference of two angles, α and β, is given by the formula cos(α - β) = cos(α)cos(β) + sin(α)sin(β). This formula is fundamental in trigonometry as it allows for the simplification of expressions involving the cosine of angle differences into products of cosines and sines.

Trigonometric Values

Trigonometric values such as cos(50°) and sin(20°) represent the ratios of the sides of a right triangle relative to its angles. Understanding these values is essential for evaluating trigonometric expressions and applying them in various contexts, including solving problems involving angles and distances.

Angle Representation

In trigonometry, angles can be represented in various forms, including degrees and radians. Converting between these forms is crucial for accurate calculations. In this context, recognizing that the expression cos(50°)cos(20°) + sin(50°)sin(20°) can be rewritten as cos(50° - 20°) demonstrates the application of the cosine difference formula.

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Related Practice

Textbook Question

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = -6i - 5j, w = -10i - 8j

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Textbook Question

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

Textbook Question

If P₁ = (-2, 3), P₂ = (-1, 5), and v is the vector from P₁ to P₂, Write v in terms of i and j.

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Textbook Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = -5j

Textbook Question

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = 3i + j

Textbook Question

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.