Ch. 7 - Applications of Trigonometry and Vectors

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 7 - Applications of Trigonometry and Vectors

Problema 9

Capitolo 8, Problema 9

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

c = 50, b = 61, C = 58°

Guida verificata passo dopo passo

Identify the given elements: side \(c = 50\), side \(b = 61\), and angle \(C = 58^\circ\). Note that angle \(C\) is opposite side \(c\).

Use the Law of Sines to find the unknown angle \(B\). The Law of Sines states: \(\frac{b}{\sin B} = \frac{c}{\sin C}\). Substitute the known values to get \(\frac{61}{\sin B} = \frac{50}{\sin 58^\circ}\).

Rearrange the equation to solve for \(\sin B\): \(\sin B = \frac{61 \times \sin 58^\circ}{50}\). Calculate this value to determine \(\sin B\) (do not find the final numeric value yet).

Analyze the value of \(\sin B\) to determine the number of possible triangles: if \(\sin B > 1\), no triangle exists; if \(\sin B = 1\), exactly one right triangle exists; if \(0 < \sin B < 1\), there may be one or two possible triangles depending on the angle \(B\) and the sum of angles.

If two triangles are possible, find the two possible values of angle \(B\) using \(B = \sin^{-1}(\sin B)\) and \(B' = 180^\circ - B\). Then check if the sum of angles \(B + C\) is less than \(180^\circ\) to confirm the validity of each triangle.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Law of Cosines

The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles. It is used to find an unknown side or angle when two sides and the included angle are known, or when all three sides are known. The formula is c² = a² + b² - 2ab cos(C), which helps determine missing elements in the triangle.

Video consigliato:

Percorso guidato

4:35

Intro to Law of Cosines

Triangle Existence and Ambiguity

Determining the number of possible triangles involves checking if the given sides and angle satisfy triangle inequality and geometric constraints. Some combinations can produce zero, one, or two triangles, especially in cases like SSA (Side-Side-Angle). Understanding these conditions helps identify if the triangle is unique or ambiguous.

Video consigliato:

Percorso guidato

9:50

Solving SSA Triangles ("Ambiguous" Case)

Angle-Side Relationships in Triangles

In any triangle, the size of an angle is directly related to the length of the opposite side. Larger angles face longer sides and vice versa. This relationship is crucial when analyzing given parts to verify if the triangle can exist and to determine the possible configurations of the triangle.

Video consigliato:

Percorso guidato

4:18

Finding Missing Side Lengths

Pratica correlata

Domanda del libro di testo

Find the length of the remaining side of each triangle. Do not use a calculator.

<IMAGE>

691

views

Domanda del libro di testo

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

〈-4, -7〉

378

views

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〉

971

views

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

557

views

Domanda del libro di testo

Consider each case and determine whether there is sufficient information to solve the triangle using the law of sines.

Three sides are known.

733

views

Domanda del libro di testo

Solve each triangle ABC that exists.

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

681