Domanda del libro di testo
Perform each calculation. See Example 3. 90° ― 51° 28'
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Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 45a
Determine whether each statement is possible or impossible. a. sec θ = ―2/3
Guida verificata passo dopo passo1
Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). This means that \(\sec \theta\) is the reciprocal of \(\cos \theta\).
Since \(\cos \theta\) ranges between \(-1\) and \(1\) for all real values of \(\theta\), the values of \(\sec \theta\) must satisfy \(|\sec \theta| \geq 1\) or be undefined where \(\cos \theta = 0\).
Check the given value \(\sec \theta = -\frac{2}{3}\). The absolute value is \(\left| -\frac{2}{3} \right| = \frac{2}{3}\), which is less than 1.
Because \(|\sec \theta|\) must be greater than or equal to 1, \(\sec \theta = -\frac{2}{3}\) is not possible for any real angle \(\theta\).
Therefore, the statement \(\sec \theta = -\frac{2}{3}\) is impossible.
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Definition and Range of Secant Function
The secant function, sec θ, is defined as the reciprocal of the cosine function: sec θ = 1/cos θ. Since cosine values range between -1 and 1, secant values are either greater than or equal to 1 or less than or equal to -1. Values between -1 and 1 are not possible for sec θ.
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Percorso guidato
6:22
Graphs of Secant and Cosecant Functions
Reciprocal Relationship Between Secant and Cosine
Because sec θ = 1/cos θ, if sec θ is given, the corresponding cosine value can be found by taking its reciprocal. This relationship helps determine if a given secant value is possible by checking if the reciprocal lies within the cosine function's valid range.
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Percorso guidato
6:22Graphs of Secant and Cosecant Functions
Determining Possibility of Trigonometric Values
To determine if a trigonometric value is possible, compare it against the function's range. For sec θ, values must satisfy |sec θ| ≥ 1. If a given value falls outside this range, it is impossible for any angle θ to produce that value.
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5:32Fundamental Trigonometric Identities
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