Domanda del libro di testo
In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 sin² x + sin x - 2 = 0
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 3, Problema 67
In Exercises 67–74, rewrite each expression in terms of the given function or functions. ;
Guida verificata passo dopo passo1
Start by rewriting the given expression clearly: \( \frac{\tan x + \cot x}{\cos x \cdot \csc x} \).
Recall the definitions of the trigonometric functions involved: \( \tan x = \frac{\sin x}{\cos x} \), \( \cot x = \frac{\cos x}{\sin x} \), and \( \csc x = \frac{1}{\sin x} \).
Rewrite the numerator \( \tan x + \cot x \) as \( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \). Find a common denominator to combine these two terms.
Rewrite the denominator \( \cos x \cdot \csc x \) as \( \cos x \cdot \frac{1}{\sin x} = \frac{\cos x}{\sin x} \).
Now, express the entire fraction as \( \frac{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}}{\frac{\cos x}{\sin x}} \). Simplify this complex fraction by multiplying numerator and denominator appropriately to eliminate the complex fraction.
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Reciprocal Trigonometric Functions
Reciprocal functions relate pairs like sine and cosecant, cosine and secant, tangent and cotangent. For example, csc x = 1/sin x and cot x = 1/tan x. Understanding these relationships helps rewrite expressions by substituting one function with its reciprocal.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Simplifying Complex Fractions
Complex fractions involve a fraction divided by another fraction or expression. Simplifying requires rewriting the numerator and denominator in terms of common functions, then multiplying by the reciprocal of the denominator to simplify the overall expression.
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Percorso guidato
4:22Dividing Complex Numbers
Expressing Trigonometric Expressions in Terms of a Given Function
This involves rewriting all parts of an expression using only the specified trigonometric function(s). For example, expressing tan x and cot x in terms of sin x and cos x, or rewriting everything in terms of cos x and csc x, to meet the problem's requirements.
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6:36Simplifying Trig Expressions
Pratica correlata
Domanda del libro di testo
In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 5 cos² x - 3 = 0
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Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin 2x = cos x
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Domanda del libro di testo
In Exercises 59–68, verify each identity.
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Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 4 cos² x = 5 - 4 sin x
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Domanda del libro di testo
In Exercises 69–74, rewrite each expression as a simplified expression containing one term. cos (α + β) cos β + sin (α + β) sin β
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