Ch. 5 - Trigonometric Identities

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

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.1.38

Capitolo 6, Problema 5.1.38

Find the remaining five trigonometric functions of θ.
cos θ = -1/4, sin θ > 0

Guida verificata passo dopo passo

Identify the quadrant where the angle \( \theta \) lies based on the given information: \( \cos \theta = -\frac{1}{4} \) and \( \sin \theta > 0 \). Since cosine is negative and sine is positive, \( \theta \) is in the second quadrant.

Use the Pythagorean identity to find \( \sin \theta \): \[ \sin^2 \theta + \cos^2 \theta = 1 \]. Substitute \( \cos \theta = -\frac{1}{4} \) to get \[ \sin^2 \theta + \left(-\frac{1}{4}\right)^2 = 1 \].

Solve for \( \sin \theta \) by isolating \( \sin^2 \theta \): \[ \sin^2 \theta = 1 - \left(-\frac{1}{4}\right)^2 = 1 - \frac{1}{16} \]. Then take the square root, remembering that \( \sin \theta > 0 \) in the second quadrant.

Calculate the remaining trigonometric functions using the definitions: \[ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta} \].

Substitute the values of \( \sin \theta \) and \( \cos \theta \) into these formulas to express \( \tan \theta \), \( \cot \theta \), \( \sec \theta \), and \( \csc \theta \) in terms of known quantities.

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Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. Given cos θ, this identity allows you to find sin θ by rearranging the equation. Since sin θ > 0, you select the positive root when solving for sin θ.

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Pythagorean Identities

Signs of Trigonometric Functions in Quadrants

The sign of sine and cosine depends on the quadrant where the angle θ lies. Since cos θ = -1/4 (negative) and sin θ > 0 (positive), θ is in the second quadrant, where sine is positive and cosine is negative. This helps determine the correct signs for all functions.

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Quadratic Formula

Definitions of the Six Trigonometric Functions

The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Knowing cos θ and sin θ allows you to calculate tangent (sin θ/cos θ), cotangent (cos θ/sin θ), secant (1/cos θ), and cosecant (1/sin θ). These definitions are essential to find all remaining functions.

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6:04

Introduction to Trigonometric Functions

Pratica correlata

Domanda del libro di testo

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

1 + cot(-θ)/cot(-θ)

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Domanda del libro di testo

Verify that each equation is an identity.

(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1

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Domanda del libro di testo

Identify the basic trigonometric function graphed, and determine whether it is even or odd.

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