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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblem 40
Chapter 8, Problem 40

Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.


(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)

Verified step by step guidance
1
Recall the Law of Sines, which states that for any triangle ABC with sides a, b, and c opposite angles A, B, and C respectively, the following holds: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\] where \(R\) is the radius of the triangle's circumscribed circle.
From the Law of Sines, express sides \(a\) and \(b\) in terms of \(\sin A\) and \(\sin B\): \[a = 2R \sin A\] \[b = 2R \sin B\]
Substitute these expressions for \(a\) and \(b\) into the left side of the equation to be proved: \[\frac{a - b}{a + b} = \frac{2R \sin A - 2R \sin B}{2R \sin A + 2R \sin B}\]
Factor out \$2R\( from numerator and denominator: \[\frac{2R (\sin A - \sin B)}{2R (\sin A + \sin B)}\] Since \)2R$ is common in numerator and denominator, it cancels out, leaving: \[\frac{\sin A - \sin B}{\sin A + \sin B}\]
This shows that \[\frac{a - b}{a + b} = \frac{\sin A - \sin B}{\sin A + \sin B}\] which completes the proof using the Law of Sines.

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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 states that in any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C. This relationship allows us to connect side lengths and angles, making it essential for proving identities involving sides and sines of angles.
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Intro to Law of Sines

Algebraic Manipulation of Ratios

Understanding how to manipulate ratios and fractions is crucial for transforming expressions like (a - b)/(a + b) and (sin A - sin B)/(sin A + sin B). This involves factoring, cross-multiplying, and simplifying terms to show equivalence between two ratios.
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Algebraic Operations on Vectors

Properties of Sine Function in Triangles

The sine function relates angles to side lengths in triangles. Recognizing how sine values change with angles and how differences and sums of sines behave helps in comparing expressions like (sin A - sin B) and (sin A + sin B), which is key to proving the given identity.
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Graph of Sine and Cosine Function
Related Practice
Textbook Question

To build the pyramids in Egypt, it is believed that giant causeways were constructed to transport the building materials to the site. One such causeway is said to have been 3000 ft long, with a slope of about 2.3°. How much force would be required to hold a 60-ton monolith on this causeway?


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Textbook Question

A balloonist is directly above a straight road 1.5 mi long that joins two villages. She finds that the town closer to her is at an angle of depression of 35°, and the farther town is at an angle of depression of 31°. How high above the ground is the balloon? 


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Textbook Question

Given vectors u and v, find: 2u.

u = 2i, v = i + j

Textbook Question

Given vectors u and v, find: 2u + 3v.

u = 2i, v = i + j

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Textbook Question

A force of 18.0 lb is required to hold a 60.0-lb stump grinder on an incline. What angle does the incline make with the horizontal?

Textbook Question

Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?