Textbook Question
Fill in the blank(s) to correctly complete each sentence.
A triangle that is not a right triangle is a(n) _________ 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 1
Use the law of sines to find the indicated part of each triangle ABC.
Find b if C = 74.2°, c = 96.3 m, B = 39.5
Verified step by step guidance1
Identify the known parts of the triangle: angle C = 74.2°, side c = 96.3 m (opposite angle C), and angle B = 39.5°. We need to find side b, which is opposite 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 and c and their opposite angles B and C.
Set up the proportion using the Law of Sines: \(\frac{b}{\sin B} = \frac{c}{\sin C}\). Substitute the known values: \(\frac{b}{\sin 39.5^\circ} = \frac{96.3}{\sin 74.2^\circ}\).
Solve for side b by multiplying both sides by \(\sin 39.5^\circ\): \(b = \frac{96.3 \times \sin 39.5^\circ}{\sin 74.2^\circ}\).
Calculate the sine values for angles 39.5° and 74.2°, then perform the multiplication and division to find the length of side 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 useful for solving triangles when given two angles and one side or two sides and a non-included angle.
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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 missing angle when two angles are known, which is essential before applying the Law of Sines if not all angles are given.
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4:47Sum and Difference of Tangent
Solving for Unknown Sides
Once the Law of Sines is set up, you can solve for an unknown side by rearranging the formula to isolate the side length. This involves substituting known angle measures and side lengths, then calculating the unknown side using algebraic manipulation and trigonometric values.
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4:18Finding Missing Side Lengths
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
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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