Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). cos 2x = cos x
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 3, Problema 72
In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x
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Identify the given expression: \( (\sec x + \csc x)(\sin x + \cos x) - 2 - \cot x \) and the function to express it in terms of: \( \tan x \).
Recall the definitions of the trigonometric functions involved: \( \sec x = \frac{1}{\cos x} \), \( \csc x = \frac{1}{\sin x} \), and \( \cot x = \frac{\cos x}{\sin x} \).
Expand the product \( (\sec x + \csc x)(\sin x + \cos x) \) by distributing each term:
\( \sec x \cdot \sin x + \sec x \cdot \cos x + \csc x \cdot \sin x + \csc x \cdot \cos x \).
Substitute the definitions into each term and simplify where possible, for example, \( \sec x \cdot \sin x = \frac{\sin x}{\cos x} = \tan x \), and \( \csc x \cdot \sin x = 1 \).
Combine all simplified terms, subtract 2 and \( \cot x \), then rewrite any remaining expressions in terms of \( \tan x \) using identities such as \( \cot x = \frac{1}{\tan x} \) to express the entire expression solely in terms of \( \tan x \).
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Reciprocal Trigonometric Functions
Reciprocal functions are the inverses of the basic sine, cosine, and tangent functions. Secant (sec x) is 1/cos x, cosecant (csc x) is 1/sin x, and cotangent (cot x) is 1/tan x. Understanding these allows rewriting expressions involving sec, csc, and cot in terms of sine, cosine, or tangent.
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6:04
Introduction to Trigonometric Functions
Algebraic Manipulation of Trigonometric Expressions
Simplifying trigonometric expressions often requires factoring, expanding, and combining like terms. Recognizing common denominators and using identities helps rewrite complex expressions in terms of specified functions, such as expressing everything in terms of tan x or sin x and cos x.
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6:36Simplifying Trig Expressions
Pythagorean and Quotient Identities
Pythagorean identities like sin²x + cos²x = 1 and quotient identities such as tan x = sin x / cos x are essential tools. They enable conversion between different trigonometric functions and simplify expressions by substituting equivalent forms.
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6:25Pythagorean Identities
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