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
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
Verified step by step guidance1
Step 1: Identify the given parts of the triangle. The problem states that sides and angles given are 'a', 'B', and 'C'. Here, 'a' is the side opposite angle 'A', and 'B' and 'C' are two angles of the triangle.
Step 2: Determine the type of information given. Since two angles (B and C) and one side (a) are known, this corresponds to the SSA (Side-Side-Angle) case, because you have one side and two angles, but the side is not between the two known angles.
Step 3: Recall the criteria for each case: SAA (two angles and a non-included side), ASA (two angles and the included side), SSA (two sides and a non-included angle), SAS (two sides and the included angle), and SSS (three sides). Since you have two angles and one side opposite an unknown angle, this is SSA.
Step 4: Decide which law to use. The Law of Sines is typically used when you have an angle-side opposite pair and another angle or side, which fits the SSA case here.
Step 5: Conclude that to begin solving the triangle with given 'a', 'B', and 'C', you should use the Law of Sines, which is given by the formula: \(\frac{a}{\sin\left( A \right)} = \frac{b}{\sin\left( B \right)} = \frac{c}{\sin\left( C \right)}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Triangle Congruence and Similarity Criteria
Triangle congruence criteria such as SAA (or AAS), ASA, SSA, SAS, and SSS describe specific combinations of sides and angles that determine a unique triangle. Understanding these helps identify what information is given and what methods apply for solving the triangle.
Recommended video:
5:35
30-60-90 Triangles
Law of Sines
The Law of Sines relates the ratios of sides to the sines of their opposite angles in any triangle. It is especially useful when two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known, allowing for the determination of unknown sides or angles.
Recommended video:
4:27Intro to Law of Sines
Law of Cosines
The Law of Cosines generalizes the Pythagorean theorem for any triangle, relating the lengths of sides to the cosine of an included angle. It is typically used when two sides and the included angle (SAS) or all three sides (SSS) are known, enabling calculation of unknown sides or angles.
Recommended video:
4:35Intro to Law of Cosines
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
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
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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