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 - 1/sec² x
725
views
Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)
Guida verificata passo dopo passo1
Recognize that the expression \( \sin\left(\frac{3\pi}{4} - x\right) \) is a sine of a difference of two angles. We can use the sine difference identity to rewrite it.
Recall the sine difference identity: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). Here, let \( A = \frac{3\pi}{4} \) and \( B = x \).
Apply the identity: \( \sin\left(\frac{3\pi}{4} - x\right) = \sin\frac{3\pi}{4} \cdot \cos x - \cos\frac{3\pi}{4} \cdot \sin x \).
Evaluate the trigonometric functions of the constant angle \( \frac{3\pi}{4} \): \( \sin\frac{3\pi}{4} \) and \( \cos\frac{3\pi}{4} \). These are standard values on the unit circle.
Substitute these values back into the expression to write \( \sin\left(\frac{3\pi}{4} - x\right) \) entirely in terms of \( \sin x \) and \( \cos x \).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Concetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Angle Difference Identity for Sine
The angle difference identity states that sin(A - B) = sin A cos B - cos A sin B. This formula allows you to express the sine of a difference of two angles as a combination of sines and cosines of the individual angles, which is essential for rewriting sin(3π/4 - x).
Video consigliato:
Percorso guidato
06:14
Sum and Difference of Sine & Cosine
Exact Values of Trigonometric Functions at Special Angles
Certain angles like π/4, π/3, and π/6 have known exact sine and cosine values. For example, sin(3π/4) = √2/2 and cos(3π/4) = -√2/2. Knowing these values helps simplify expressions involving these angles without using a calculator.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
Function Notation and Variable Isolation
Expressing a trigonometric function in terms of θ or x alone means rewriting the expression so that the variable appears only inside the trigonometric functions, not combined with constants. This often involves applying identities to separate constants and variables clearly.
Video consigliato:
Percorso guidato
06:01i & j Notation
Pratica correlata
Domanda del libro di testo
Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α
830
views
Domanda del libro di testo
Verify that each equation is an identity.
(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)
787
views
Domanda del libro di testo
Find sinθ.
sec θ = 7/2, tan θ < 0
33
views
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 cos θ
788
views
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 α sin α
740
views