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Ch. 2 - Acute Angles and Right Triangles
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. 2 - Acute Angles and Right TrianglesProblema 2.3.36
Capitolo 3, Problema 2.3.36

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.
sec θ = 1.1606249

Guida verificata passo dopo passo
1
Recall the definition of secant in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\). This means that \(\cos \theta = \frac{1}{\sec \theta}\).
Substitute the given value of \(\sec \theta\) into the equation: \(\cos \theta = \frac{1}{1.1606249}\).
Calculate the value of \(\cos \theta\) using the reciprocal of the given secant value (do not compute the final number here, just set up the expression).
Use the inverse cosine function to find \(\theta\): \(\theta = \cos^{-1}(\text{value from step 3})\). This will give \(\theta\) in radians or degrees depending on your calculator settings.
Since the problem asks for \(\theta\) in degrees within the interval \([0^\circ, 90^\circ)\), ensure your calculator is set to degrees and that the angle you find lies within this interval.

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Definition of Secant Function

The secant function, sec θ, is the reciprocal of the cosine function, defined as sec θ = 1/cos θ. Understanding this relationship allows you to convert the given secant value into a cosine value, which is often easier to work with when solving for θ.
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Graphs of Secant and Cosecant Functions

Inverse Trigonometric Functions

To find the angle θ from a trigonometric value, you use inverse functions such as arccos or arcsec. Since sec θ = 1.1606249, you first find cos θ = 1/1.1606249, then apply the inverse cosine function to determine θ within the specified interval.
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Introduction to Inverse Trig Functions

Domain and Range Restrictions

The problem restricts θ to the interval [0°, 90°), which corresponds to the first quadrant where cosine values are positive. This restriction ensures a unique solution for θ and guides the selection of the correct angle from the inverse function's output.
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Domain and Range of Function Transformations
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