Textbook Question
In Exercises 9–16, letu = 2i - j, v = 3i + j, and w = i + 4j.Find each specified scalar.u ⋅ (v + w)
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 8
In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
Verified step by step guidance1
Identify the type of triangle problem you are dealing with: whether you have two sides and an included angle (SAS), two angles and a side (ASA or AAS), or three sides (SSS). This will determine which trigonometric laws or formulas to use.
If you have two sides and the included angle (SAS), use the Law of Cosines to find the third side: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] where \(a\) and \(b\) are known sides and \(C\) is the included angle.
Once you have all three sides, or if you start with two angles and a side (ASA or AAS), use the Law of Sines to find unknown sides or angles: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] where \(a\), \(b\), \(c\) are sides opposite angles \(A\), \(B\), \(C\) respectively.
If you have all three sides (SSS), use the Law of Cosines to find one angle first, then use the Law of Sines or Law of Cosines again to find the remaining angles.
After finding all sides and angles, round the side lengths to the nearest tenth and the angle measures to the nearest degree as required.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Types of Triangles and Their Properties
Understanding whether a triangle is right, acute, or obtuse is essential because it determines which trigonometric rules apply. Recognizing side lengths and angle measures helps in selecting appropriate methods for solving the triangle.
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4:42
Review of Triangles
Law of Sines and Law of Cosines
These laws are fundamental for solving triangles when not all sides and angles are known. The Law of Sines relates ratios of sides to the sines of opposite angles, while the Law of Cosines generalizes the Pythagorean theorem for any triangle.
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4:35Intro to Law of Cosines
Rounding and Approximation in Trigonometry
Rounding side lengths to the nearest tenth and angles to the nearest degree ensures practical and interpretable results. It is important to apply consistent rounding rules after calculations to maintain accuracy and clarity.
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5:32Fundamental Trigonometric Identities
Related Practice
Textbook Question
In Exercises 5–12, sketch each vector as a position vector and find its magnitude.v = -6i - 2j
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Textbook Question
In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i, w = j
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Textbook Question
In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.a = 5, b = 7, C = 42°
Textbook Question
In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle.A = 162°, b = 11.2, c = 48.2
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Textbook Question
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. projᵥᵥv