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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 4 - Laws of Sines and Cosines; VectorsProblem 4
Chapter 4, Problem 4

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 guidance
1
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, use the Law of Sines to find unknown angles or sides: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] where \(a\), \(b\), \(c\) are sides opposite angles \(A\), \(B\), \(C\) respectively.
Calculate the remaining angles by using the fact that the sum of angles in a triangle is 180 degrees: \[ A + B + C = 180^\circ \] This helps find the last unknown angle once two are known.
Round all side lengths to the nearest tenth and all angle measures to the nearest degree as the final step to complete the solution.

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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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Review of Triangles

Law of Sines and Law of Cosines

These laws relate the sides and angles of any triangle, enabling the calculation of unknown measures. The Law of Sines is useful when given two angles and one side or two sides and a non-included angle, while the Law of Cosines applies when two sides and the included angle or all three sides are known.
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Intro to Law of Cosines

Rounding and Angle Measurement Conventions

Accurate rounding of side lengths to the nearest tenth and angles to the nearest degree ensures clarity and precision in answers. Understanding degree measurement and proper rounding rules is crucial for presenting final solutions correctly.
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Drawing Angles in Standard Position
Related Practice
Textbook Question

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

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Textbook Question

Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.


P₁ = (2, -5), P₂ = (-6, 6)

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Textbook Question

Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.


P₁ = (-1, 6), P₂ = (7, -5)

Textbook Question

In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.

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Textbook Question

The magnitude and direction angle of v are ||v|| = 12 and θ = 60°. Express v in terms of i and j.

Textbook Question

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

v = -5j