Textbook Question
Fill in the blank(s) to correctly complete each sentence.
A triangle that is not a right triangle is a(n) _________ triangle.
1
views
Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
Not the one you use?Change textbookSelect a different textbook
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 3
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
Verified step by step guidance1
Identify the known parts of the triangle: angle \(C = 51.3^\circ\), side \(c = 68.3\) m (opposite angle \(C\)), and side \(b = 58.2\) m (opposite angle \(B\)). We need to find angle \(B\).
Recall the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). We will use the relationship between sides \(b\), \(c\) and angles \(B\), \(C\).
Set up the equation using the Law of Sines for sides \(b\) and \(c\): \(\frac{b}{\sin B} = \frac{c}{\sin C}\). Substitute the known values: \(\frac{58.2}{\sin B} = \frac{68.3}{\sin 51.3^\circ}\).
Solve for \(\sin B\) by cross-multiplying: \(\sin B = \frac{58.2 \times \sin 51.3^\circ}{68.3}\). Calculate the right side to find \(\sin B\) (do not compute the final value here).
Find angle \(B\) by taking the inverse sine (arcsin) of the value obtained for \(\sin B\): \(B = \sin^{-1}(\sin B)\). Remember to consider the possible ambiguous case for angle \(B\) in the triangle.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Was this helpful?
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 ratios of the lengths of sides of a triangle to the sines of their opposite angles. It states that (a/sin A) = (b/sin B) = (c/sin C). This law is useful for solving triangles when given two angles and one side or two sides and a non-included angle.
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°. Knowing two angles allows you to find the third by subtracting their sum from 180°. This property helps verify or find missing angles after applying the Law of Sines.
Recommended video:
4:47Sum and Difference of Tangent
Solving for an Unknown Angle
When two sides and an angle opposite one of them are known, the Law of Sines can be used to find an unknown angle by rearranging the formula and using inverse sine functions. Care must be taken to consider the ambiguous case where two different angles may satisfy the equation.
Recommended video:
2:28Solving Problems with Complementary & Supplementary Angles
Related Practice
Textbook Question
Which one of the following sets of data does not determine a unique triangle?
a. A = 50°, b = 21, a = 19
b. A = 45°, b = 10, a = 12
c. A = 130°, b = 4, a = 7
d. A = 30°, b = 8, a = 4
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
<IMAGE>
3
views
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
<IMAGE>
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
<IMAGE>
3
views
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
1
views