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 28

Capitolo 3, Problema 28

Write each function in terms of its cofunction. Assume all angles involved are acute angles. See Example 2. cos(θ + 20°)

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Recall the cofunction identities for acute angles, which relate trigonometric functions of complementary angles: for example, \(\cos(\alpha) = \sin(90^\circ - \alpha)\) and \(\sin(\alpha) = \cos(90^\circ - \alpha)\).

Identify the function you want to rewrite in terms of its cofunction. Here, the function is \(\cos(\theta + 20^\circ)\).

Apply the cofunction identity for cosine: replace \(\cos(\alpha)\) with \(\sin(90^\circ - \alpha)\). In this case, \(\alpha = \theta + 20^\circ\).

Write the expression as \(\sin\left(90^\circ - (\theta + 20^\circ)\right)\), which simplifies the argument inside the sine function.

Simplify the angle inside the sine function to get \(\sin(90^\circ - \theta - 20^\circ)\), which further simplifies to \(\sin(70^\circ - \theta)\).

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Cofunction Identities

Cofunction identities relate pairs of trigonometric functions such that the function of an angle equals the cofunction of its complement. For example, sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ). These identities are essential for rewriting functions in terms of their cofunctions.

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Cofunction Identities

Angle Sum in Trigonometric Functions

The angle sum formula allows the evaluation of trigonometric functions of sums of angles, such as cos(θ + 20°) = cos θ cos 20° - sin θ sin 20°. Understanding this helps in breaking down complex angles into simpler components for manipulation or substitution.

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Introduction to Trigonometric Functions

Acute Angles and Complementary Angles

Since all angles are acute (less than 90°), the complement of an angle (90° - θ) is also acute. This ensures the validity of cofunction identities and simplifies the process of expressing functions in terms of their cofunctions without ambiguity.

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Intro to Complementary & Supplementary Angles

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