Domanda del libro di testo
Factor each trigonometric expression.
cot⁴ x + 3 cot² x + 2
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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 5.42
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
Guida verificata passo dopo passo1
Recognize that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \).
Rewrite \( \frac{1}{\tan^2 \alpha} \) as \( \cot^2 \alpha \) using the identity \( \cot \alpha = \frac{1}{\tan \alpha} \).
Substitute \( \cot^2 \alpha \) for \( \frac{1}{\tan^2 \alpha} \) in the expression.
Simplify \( \cot \alpha \tan \alpha \) to 1, since \( \cot \alpha = \frac{1}{\tan \alpha} \).
Combine the simplified terms: \( \cot^2 \alpha + 1 \).
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Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Fundamental identities, such as the Pythagorean identities, reciprocal identities, and quotient identities, are essential for simplifying trigonometric expressions. Understanding these identities allows students to manipulate and transform expressions effectively.
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Percorso guidato
5:32
Fundamental Trigonometric Identities
Reciprocal Functions
Reciprocal functions in trigonometry refer to the relationships between sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent. For example, the tangent function is the reciprocal of cotangent, and this relationship is crucial when simplifying expressions. Recognizing these relationships helps in rewriting expressions in a more manageable form.
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Percorso guidato
3:23Secant, Cosecant, & Cotangent on the Unit Circle
Simplification Techniques
Simplification techniques in trigonometry involve rewriting complex expressions into simpler forms using identities and algebraic manipulation. This may include factoring, combining like terms, or substituting equivalent expressions. Mastering these techniques is vital for solving trigonometric equations and understanding their behavior in various contexts.
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Percorso guidato
6:20Example 6
Pratica correlata
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
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Domanda del libro di testo
Factor each trigonometric expression.
sec² θ - 1
1013
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Domanda del libro di testo
Factor each trigonometric expression.
sin³ α + cos³ α
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Domanda del libro di testo
Verify that each equation is an identity.
tan² α sin² α = tan² α + cos² α - 1
808
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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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