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Solving Trigonometric Equations
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
sin² θ = cos² θ
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 1, Problema 1.1.60
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
sin x²
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Understand the definitions: A function f(x) is even if f(-x) = f(x) for all x in the domain, and odd if f(-x) = -f(x) for all x in the domain.
Consider the function given: f(x) = sin(x²). We need to check the behavior of f(-x) compared to f(x).
Calculate f(-x): Substitute -x into the function to get f(-x) = sin((-x)²). Since (-x)² = x², this simplifies to sin(x²).
Compare f(-x) and f(x): We find that f(-x) = sin(x²) = f(x), which matches the condition for an even function.
Conclude: Since f(-x) = f(x), the function sin(x²) is even. It does not satisfy the condition for an odd function, which would require f(-x) = -f(x).
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Even Functions
A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the y-axis. Common examples include polynomial functions with only even powers, such as f(x) = x².
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6:13
Exponential Functions
Odd Functions
A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of the function is symmetric with respect to the origin. Typical examples include polynomial functions with only odd powers, such as f(x) = x³.
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Percorso guidato
06:21Properties of Functions
Trigonometric Functions
Trigonometric functions, such as sine and cosine, have specific properties regarding their symmetry. The sine function is an odd function, meaning sin(-x) = -sin(x). Understanding these properties is crucial when analyzing functions like sin(x²) to determine their evenness or oddness.
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6:04Introduction to Trigonometric Functions
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Domanda del libro di testo
Increasing and Decreasing Functions
Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
y = −x³
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Domanda del libro di testo
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
sin (A − B) = sin A cos B − cos A sin B
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Domanda del libro di testo
Graph the functions in Exercises 37–56.
y = (x + 1)²/³
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Domanda del libro di testo
Shifting Graphs
Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.
y = x³ Left 1, down 1
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Domanda del libro di testo
General Sine Curves
For
f(x) = A sin ((2π/B)(x – C) +D
identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.
y = ½ sin (πx – x) + ½
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