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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 34
Capitolo 8, Problema 34

Apply the law of sines to the following:


A = 104°, a = 26.8, b = 31.3.


What happens when we try to find the measure of angle B using a calculator?

Guida verificata passo dopo passo
1
Recall the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B}\), where \(A\) and \(B\) are angles opposite sides \(a\) and \(b\) respectively.
Substitute the known values into the formula: \(\frac{26.8}{\sin 104^\circ} = \frac{31.3}{\sin B}\).
Rearrange the equation to solve for \(\sin B\): \(\sin B = \frac{31.3 \times \sin 104^\circ}{26.8}\).
Use a calculator to evaluate the right-hand side expression to find the numerical value of \(\sin B\).
Observe the result: if the value of \(\sin B\) is greater than 1, this indicates that no angle \(B\) exists with such a sine value, meaning the triangle cannot be formed with the given measurements.

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Law of Sines

The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of their opposite angles: (a/sin A) = (b/sin B) = (c/sin C). It is used to find unknown sides or angles in any triangle when given sufficient information, especially in non-right triangles.
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Percorso guidato
4:27
Intro to Law of Sines

Ambiguous Case in the Law of Sines

When using the Law of Sines to find an angle, the inverse sine function can yield two possible angle measures (an acute and an obtuse angle) because sin(θ) = sin(180° - θ). This ambiguity can cause confusion or multiple solutions, especially in SSA (Side-Side-Angle) configurations.
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Percorso guidato
9:50
Solving SSA Triangles ("Ambiguous" Case)

Using a Calculator for Inverse Sine

Calculators typically return the principal value of the inverse sine function, which is an angle between -90° and 90°. When solving triangles, this means the calculator gives only one possible angle, so you must consider the possibility of a second valid angle (180° minus the calculator’s result) to fully solve the problem.
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Percorso guidato
4:03
Inverse Sine
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