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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 7.19
Capitolo 8, Problema 7.19

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


A = 142.13°, b = 5.432 ft, a = 7.297 ft

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Step 1: Use the Law of Sines to find angle B. The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Substitute the known values: \( \frac{7.297}{\sin 142.13^\circ} = \frac{5.432}{\sin B} \).
Step 2: Solve for \( \sin B \) by rearranging the equation: \( \sin B = \frac{5.432 \cdot \sin 142.13^\circ}{7.297} \).
Step 3: Calculate \( \sin B \) using the value obtained from Step 2.
Step 4: Determine angle B by taking the inverse sine (arcsin) of \( \sin B \).
Step 5: Find angle C using the fact that the sum of angles in a triangle is 180°. Use the equation: \( C = 180^\circ - A - B \).

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Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. This fundamental principle allows us to find unknown angles when at least one angle is known. In this case, knowing angle A enables us to calculate the other two angles by subtracting A from 180°.
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Solving Right Triangles with the Pythagorean Theorem

Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It is expressed as a/b = sin(A)/sin(B) = c/sin(C). This law is particularly useful in non-right triangles, allowing us to find unknown angles or sides when we have sufficient information, such as two sides and a non-included angle.
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Intro to Law of Sines

Angle of Elevation and Depression

The angle of elevation is the angle formed by the line of sight when looking up from a horizontal line, while the angle of depression is formed when looking down. Understanding these angles is crucial in trigonometry, especially in real-world applications involving heights and distances, as they often require the use of trigonometric ratios to solve for unknown values.
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Coterminal Angles
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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.



2c

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Solve each triangle. See Examples 2 and 3.


C = 45.6°, b = 8.94 m, a = 7.23 m

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


a = 50, b = 26, A = 95°

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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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