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 5.1.36

Capitolo 6, Problema 5.1.36

Find the remaining five trigonometric functions of θ.
sin θ = -4/5, cos θ < 0

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Identify the given information: \(\sin \theta = -\frac{4}{5}\) and \(\cos \theta < 0\). This tells us the sine value and the quadrant where \(\theta\) lies. Since sine is negative and cosine is negative, \(\theta\) is in the third quadrant.

Recall the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Use this to find \(\cos \theta\) by substituting the known sine value: \(\left(-\frac{4}{5}\right)^2 + \cos^2 \theta = 1\).

Calculate \(\cos^2 \theta\) from the equation: \(\cos^2 \theta = 1 - \left(-\frac{4}{5}\right)^2\). Then take the square root to find \(\cos \theta\). Remember to choose the negative root because \(\cos \theta < 0\) in the third quadrant.

Find \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) you have found.

Calculate the remaining three trigonometric functions using their reciprocal relationships: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Substitute the known values to express each function.

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Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. Given sin θ, this identity allows you to find cos θ by rearranging the equation. It is fundamental for determining unknown trigonometric functions when one function value is known.

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6:25

Pythagorean Identities

Sign of Trigonometric Functions in Quadrants

The sign of sine, cosine, and other trig functions depends on the quadrant where the angle θ lies. Since sin θ = -4/5 (negative) and cos θ < 0 (also negative), θ is in the third quadrant, where both sine and cosine are negative. This helps determine the correct sign of the functions.

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6:36

Quadratic Formula

Definitions of the Six Trigonometric Functions

The six trig functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios of sides in a right triangle or coordinates on the unit circle. Knowing sin θ and cos θ allows calculation of the others using their definitions, such as tan θ = sin θ / cos θ and sec θ = 1 / cos θ.

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6:04

Introduction to Trigonometric Functions

Pratica correlata

Domanda del libro di testo

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°

789

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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.

tan(-θ)/sec θ

716

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Domanda del libro di testo

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

sec x/csc x

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

570