Domanda del libro di testo
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1/ tan² α + cot α tan α
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 5.24
Factor each trigonometric expression.
sec² θ - 1
Guida verificata passo dopo passo1
Recognize that the expression \( \sec^2 \theta - 1 \) is a difference of squares.
Recall the Pythagorean identity: \( \sec^2 \theta = 1 + \tan^2 \theta \).
Substitute \( \sec^2 \theta \) with \( 1 + \tan^2 \theta \) in the expression: \( (1 + \tan^2 \theta) - 1 \).
Simplify the expression: \( \tan^2 \theta \).
Factor \( \tan^2 \theta \) as \( (\tan \theta)(\tan \theta) \).
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Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One of the fundamental identities is the Pythagorean identity, which states that sec² θ = 1 + tan² θ. Understanding these identities is crucial for simplifying and factoring trigonometric expressions.
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Percorso guidato
5:32
Fundamental Trigonometric Identities
Factoring Techniques
Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. Common techniques include recognizing patterns such as the difference of squares, which applies to expressions like sec² θ - 1. Mastery of these techniques is essential for solving trigonometric equations efficiently.
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Percorso guidato
6:08Factoring
Difference of Squares
The difference of squares is a specific algebraic identity that states a² - b² = (a - b)(a + b). This concept is particularly useful in factoring expressions where one term is the square of a variable or function. In the case of sec² θ - 1, it can be factored as (sec θ - 1)(sec θ + 1), illustrating the application of this identity in trigonometry.
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Percorso guidato
4:47Sum and Difference of Tangent
Pratica correlata
Domanda del libro di testo
Factor each trigonometric expression.
cot⁴ x + 3 cot² x + 2
724
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Domanda del libro di testo
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
cot t tan t
721
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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.
-sec² (-θ) + sin² (-θ) + cos² (-θ)
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Domanda del libro di testo
Match each expression in Column I with its value in Column II.
10. cos 67.5°
729
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Domanda del libro di testo
Simplify each expression.
± √[(1 + cos (x/4))/2]
726
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