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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 23
Capitolo 3, Problema 23

Evaluate each expression. Give exact values. sec² 300° - 2 cos² 150°

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Recall the definitions and identities involved: \( \sec \theta = \frac{1}{\cos \theta} \) and therefore \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \). Also, remember the Pythagorean identity \( \sec^2 \theta = 1 + \tan^2 \theta \), but here it is easier to work directly with cosine values.
Evaluate \( \cos 300^\circ \) and \( \cos 150^\circ \) using the unit circle or reference angles. For \( 300^\circ \), find the cosine value in the fourth quadrant, and for \( 150^\circ \), find the cosine value in the second quadrant.
Square the cosine values to get \( \cos^2 300^\circ \) and \( \cos^2 150^\circ \). Then compute \( \sec^2 300^\circ = \frac{1}{\cos^2 300^\circ} \).
Substitute these values into the expression \( \sec^2 300^\circ - 2 \cos^2 150^\circ \) and simplify the expression step-by-step, keeping the exact values (like fractions or square roots) without converting to decimals.
Combine like terms and simplify the expression fully to get the exact value of the original expression.

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Trigonometric Functions and Their Values

Understanding the basic trigonometric functions such as cosine and secant is essential. Secant is the reciprocal of cosine, so sec²θ = 1/cos²θ. Knowing how to find exact values of these functions at specific angles, especially those related to special angles and their positions on the unit circle, is crucial.
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Introduction to Trigonometric Functions

Unit Circle and Angle Reference

The unit circle helps determine the exact values of trigonometric functions for angles like 150° and 300°. Recognizing the quadrant in which the angle lies allows you to assign the correct sign to the function values. For example, 150° is in the second quadrant where cosine is negative, and 300° is in the fourth quadrant where cosine is positive.
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Reference Angles on the Unit Circle

Pythagorean Identities and Simplification

Using identities such as sec²θ = 1 + tan²θ or the Pythagorean identity sin²θ + cos²θ = 1 can simplify expressions. In this problem, rewriting sec² 300° in terms of cosine and combining like terms helps in evaluating the expression exactly. Simplification skills are key to obtaining the final exact value.
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Pythagorean Identities
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