Textbook Question
In Exercises 22–24, sketch each vector as a position vector and find its magnitude.v = -3i - 4j
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 23
In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.
a = 63, b = 22, c = 50
Verified step by step guidance1
Identify the given elements of the triangle: sides \(a = 63\), \(b = 22\), and \(c = 50\). Since all three sides are known, this is a Side-Side-Side (SSS) case.
Use the Law of Cosines to find one of the angles. For example, to find angle \(A\) opposite side \(a\), use the formula:
\[\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\]
Calculate \(\cos(A)\) by substituting the known side lengths into the formula, then find angle \(A\) by taking the inverse cosine (arccos) of that value:
\[A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right)\]
Once angle \(A\) is found, use the Law of Cosines again to find another angle, for example angle \(B\) opposite side \(b\):
\[\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}\]
Then find \(B\) by taking the inverse cosine.
Finally, find the third angle \(C\) using the fact that the sum of angles in a triangle is \(180^\circ\):
\[C = 180^\circ - A - B\]
Round all angles to the nearest degree and all sides are already given.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Triangle Classification and Properties
Understanding the types of triangles (scalene, isosceles, equilateral) and their properties is essential. Given three sides, the triangle inequality theorem must be checked to ensure a valid triangle exists before solving for angles.
Recommended video:
4:42
Review of Triangles
Law of Cosines
The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles. It is used to find unknown angles when all three sides are known, using the formula: c² = a² + b² - 2ab cos(C).
Recommended video:
4:35Intro to Law of Cosines
Rounding and Angle Measurement
After calculating side lengths and angles, results must be rounded appropriately. Lengths are rounded to the nearest tenth, and angles to the nearest degree, ensuring answers are precise and practical for real-world applications.
Recommended video:
5:31Reference Angles on the Unit Circle
Related Practice
Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = i + j, w = i - j
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
u - v
Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 2i + 8j, w = 4i - j
Textbook Question
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 10, b = 30, A = 150°
Textbook Question
In Exercises 22–24, sketch each vector as a position vector and find its magnitude.
v = -3j