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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblema 7.33
Capitolo 8, Problema 7.33

To find the distance AB across a river, a surveyor laid off a distance BC = 354 m on one side of the river. It is found that B = 112° 10' and C = 15° 20'. Find AB. See the figure.


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Guida verificata passo dopo passo
1
Identify the triangle involved: points A, B, and C form a triangle where BC = 354 m is known, and angles at B and C are given as 112° 10' and 15° 20' respectively.
Convert the given angles from degrees and minutes to decimal degrees for easier calculation: for example, 112° 10' = 112 + 10/60 degrees.
Calculate the third angle at point A using the triangle angle sum property: \(A = 180^\circ - B - C\).
Use the Law of Sines to relate the sides and angles in the triangle: \(\frac{AB}{\sin C} = \frac{BC}{\sin A}\).
Rearrange the Law of Sines formula to solve for AB: \(AB = \frac{BC \times \sin C}{\sin A}\). Substitute the known values of BC, \(\sin C\), and \(\sin A\) to find AB.

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Law of Sines

The Law of Sines relates the sides and angles of any triangle, stating that the ratio of a side length to the sine of its opposite angle is constant. It is essential for solving triangles when two angles and one side or two sides and one angle are known, enabling calculation of unknown sides or angles.
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Intro to Law of Sines

Angle Measurement and Conversion

Understanding how to interpret and convert angles given in degrees and minutes is crucial. Since angles are often provided in degrees (°) and minutes ('), converting these to decimal degrees or radians facilitates accurate calculations in trigonometric formulas.
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Reference Angles on the Unit Circle

Triangle Geometry in Surveying

Surveying problems often involve forming triangles between points on the ground and using measured distances and angles to find unknown lengths. Recognizing how to apply trigonometric principles to real-world layouts, such as across a river, is key to solving for distances like AB.
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05:29
Adding Vectors Geometrically
Pratica correlata
Domanda del libro di testo

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.



2c

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Domanda del libro di testo

Solve each triangle. See Examples 2 and 3.


C = 45.6°, b = 8.94 m, a = 7.23 m

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Domanda del libro di testo

Determine the number of triangles ABC possible with the given parts.


a = 31, b = 26, B = 48°

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Domanda del libro di testo

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈5, 7〉

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Domanda del libro di testo

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.


a + b

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Domanda del libro di testo

Solve each triangle ABC that exists.

A = 96.80°, b = 3.589 ft, a = 5.818 ft

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