Textbook Question
Use the law of sines to find the indicated part of each triangle ABC.
Find B if C = 51.3°, c = 68.3 m, b = 58.2 m
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 3c
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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Verified step by step guidance1
Identify the given point's coordinates and the angle it makes with the positive x-axis, if provided, or determine the vertical distance from the point to the x-axis.
Recall that the line segment of length \(L\) is drawn from the point to the positive x-axis, forming a triangle with the x-axis and the segment from the origin to the foot of the perpendicular.
Understand that a triangle can be formed only if the length \(L\) is greater than the shortest distance from the point to the x-axis; if \(L\) is less than this distance, no triangle can be formed.
Express the shortest distance from the point to the x-axis mathematically, which is the absolute value of the y-coordinate of the point, say \(d = |y|\).
Conclude that for no triangle to be formed, the length \(L\) must satisfy \(L < d\), meaning the segment is too short to reach the x-axis and form a triangle.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Triangle Inequality Theorem
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. This principle helps determine whether a triangle can be formed given certain side lengths, ensuring the segments can connect to form a closed shape.
Recommended video:
5:19
Solving Right Triangles with the Pythagorean Theorem
Distance from a Point to the x-axis
The distance from a point to the x-axis is the absolute value of the point's y-coordinate. This distance is crucial when drawing a segment from the point to the x-axis, as it sets a minimum length for the segment and influences the possible lengths L that can form a triangle.
Recommended video:
6:58Convert Points from Rectangular to Polar
Conditions for No Triangle Formation
No triangle is formed when the segment length L violates the triangle inequality, such as being too short or too long relative to other sides. Understanding these conditions helps identify values of L for which a triangle cannot exist, often involving equality or impossible side length combinations.
Recommended video:
6:08Evaluating Sums and Differences Given Conditions
Related Practice
Textbook Question
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°
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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
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?
a. two triangles
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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
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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