Ch. 1 - Angles and the Trigonometric Functions

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Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problema 29

Capitolo 1, Problema 29

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sec² 23° - tan² 23°

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Recall the Pythagorean identity involving secant and tangent: \(\sec^{2} \theta - \tan^{2} \theta = 1\).

Identify the angle in the problem: here, \(\theta = 23^\circ\).

Apply the identity directly by substituting \(\theta = 23^\circ\) into the expression: \(\sec^{2} 23^\circ - \tan^{2} 23^\circ\).

Since the identity holds for all angles where these functions are defined, the expression simplifies to 1 without further calculation.

Therefore, the value of \(\sec^{2} 23^\circ - \tan^{2} 23^\circ\) is 1 by the Pythagorean identity.

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Pythagorean Identity for Secant and Tangent

The identity sec²θ - tan²θ = 1 is a fundamental Pythagorean identity in trigonometry. It relates the secant and tangent functions of the same angle and allows simplification of expressions without a calculator.

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Pythagorean Identities

Definition of Secant and Tangent Functions

Secant (sec θ) is the reciprocal of cosine (1/cos θ), and tangent (tan θ) is the ratio of sine to cosine (sin θ/cos θ). Understanding these definitions helps in applying identities and simplifying trigonometric expressions.

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Graphs of Secant and Cosecant Functions

Using Identities to Simplify Expressions

Trigonometric identities allow rewriting complex expressions into simpler forms. Recognizing which identity applies enables solving problems efficiently without numerical approximation or calculators.

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Simplifying Trig Expressions

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