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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 5.1.38
Chapter 6, Problem 5.1.38

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

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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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Related Practice
Textbook Question

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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Textbook Question

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


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Textbook Question

Use identities to correctly complete each sentence.


If sin θ = ⅔, then -sin(-θ) = ________________.

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Textbook Question

Factor each trigonometric expression.

(tan x + cot x)² - (tan x - cot x)²

Textbook Question

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

csc 18°

Textbook Question

Verify that each equation is an identity.

tan² α sin² α = tan² α + cos² α - 1