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 5

Capitolo 8, Problema 5

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

a = 50, b = 26, A = 95°

Guida verificata passo dopo passo

Identify the given elements: side \(a = 50\), side \(b = 26\), and angle \(A = 95^\circ\). Note that angle \(A\) is opposite side \(a\).

Use the Law of Sines to find angle \(B\). The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Substitute the known values to get \(\frac{50}{\sin 95^\circ} = \frac{26}{\sin B}\).

Rearrange the equation to solve for \(\sin B\): \(\sin B = \frac{26 \times \sin 95^\circ}{50}\).

Evaluate \(\sin B\) and determine the possible values of angle \(B\). Since \(\sin B\) can correspond to two different angles between \(0^\circ\) and \(180^\circ\) (one acute and one obtuse), consider both possibilities for \(B\).

For each possible value of \(B\), calculate angle \(C\) using the triangle angle sum property: \(C = 180^\circ - A - B\). Check if \(C\) is positive and valid. The number of valid triangles corresponds to the number of valid \(C\) values found.

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Law of Sines

The Law of Sines relates the sides and angles of a triangle through the ratio a/sin A = b/sin B = c/sin C. It is essential for solving triangles when given two sides and an angle, especially to find unknown angles or sides.

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Percorso guidato

4:27

Intro to Law of Sines

Ambiguous Case of SSA Triangles

When two sides and a non-included angle (SSA) are given, there can be zero, one, or two possible triangles. This ambiguity arises because the given angle and side lengths may correspond to different triangle configurations.

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Percorso guidato

9:50

Solving SSA Triangles ("Ambiguous" Case)

Triangle Inequality and Angle-Side Relationships

The triangle inequality states that the sum of any two sides must be greater than the third. Also, larger angles face longer sides. These principles help determine if a triangle is possible and how many distinct triangles can be formed.

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Percorso guidato

5:35

30-60-90 Triangles

Pratica correlata

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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.

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Use the law of sines to find the indicated part of each triangle ABC.

Find b if a = 165 m, A = 100.2°, B = 25.0°

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

a = 31, b = 26, B = 48°

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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.

-b

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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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Domanda del libro di testo

Find the unknown angles in triangle ABC for each triangle that exists.