Textbook Question
In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?
c. no triangle
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Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 8, Problem 4
Use the law of sines to find the indicated part of each triangle ABC.
Find b if a = 165 m, A = 100.2°, B = 25.0°
Verified step by step guidance1
Identify the known values from the problem: side \(a = 165\) m, angle \(A = 100.2^\circ\), and angle \(B = 25.0^\circ\).
Use the fact that the sum of angles in a triangle is \(180^\circ\) to find angle \(C\): calculate \(C = 180^\circ - A - B\).
Write down the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B}\).
Rearrange the formula to solve for side \(b\): \(b = \frac{a \cdot \sin B}{\sin A}\).
Substitute the known values of \(a\), \(A\), and \(B\) into the formula and prepare to calculate \(b\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Law of Sines
The Law of Sines relates the sides and angles of a triangle by stating that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides. It is expressed as (a/sin A) = (b/sin B) = (c/sin C), and is especially useful for solving triangles when two angles and one side or two sides and a non-included angle are known.
Recommended video:
4:27
Intro to Law of Sines
Triangle Angle Sum Property
The sum of the interior angles in any triangle is always 180°. This property allows you to find the third angle when two angles are known, which is essential for applying the Law of Sines correctly, as all three angles must be known or determined to solve for unknown sides.
Recommended video:
4:47Sum and Difference of Tangent
Solving for Unknown Sides
Once the Law of Sines is set up, solving for an unknown side involves rearranging the formula to isolate the desired side length. This requires substituting known values for sides and angles, calculating sine values, and performing algebraic manipulation to find the missing side accurately.
Recommended video:
4:18Finding Missing Side Lengths
Related Practice
Textbook Question
Determine the number of triangles ABC possible with the given parts.
a = 50, b = 26, A = 95°
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Textbook Question
CONCEPT PREVIEW Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.
-b
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Textbook Question
CONCEPT PREVIEW Assume a triangle ABC has standard labeling.
a. Determine whether SAA, ASA, SSA, SAS, or SSS is given.
b. Determine whether the law of sines or the law of cosines should be used to begin solving the triangle.
a, B, and C
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Textbook Question
In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?
b. exactly one triangle
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592
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Textbook Question
Find the unknown angles in triangle ABC for each triangle that exists.
A = 142.13°, b = 5.432 ft, a = 7.297 ft
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