Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problema 56

Capitolo 3, Problema 56

Give the exact value of each expression. See Example 5. sec 45°

Guida verificata passo dopo passo

Recall the definition of secant in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).

Identify the angle given: \(45^\circ\).

Find the exact value of \(\cos 45^\circ\). From the unit circle or special triangles, \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).

Substitute the value of \(\cos 45^\circ\) into the secant formula: \(\sec 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}}\).

Simplify the expression by multiplying numerator and denominator appropriately to find the exact value of \(\sec 45^\circ\).

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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). It is important to understand this relationship to find the exact value of secant for a given angle.

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Graphs of Secant and Cosecant Functions

Exact Values of Trigonometric Functions for Special Angles

Certain angles like 30°, 45°, and 60° have well-known exact trigonometric values. For 45°, cos(45°) = √2/2, which helps in calculating sec(45°) precisely without a calculator.

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Introduction to Trigonometric Functions

Simplifying Radical Expressions

After finding the reciprocal of cosine, simplifying the resulting expression, often involving square roots, is necessary to express the answer in its simplest exact form, such as rationalizing denominators.

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Simplifying Trig Expressions

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