Domanda del libro di testo
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
h(t) = |t³|
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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 1.3.48
Using the Half-Angle Formulas
Find the function values in Exercises 47–50.
cos² 5π/12
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First, recognize that the expression cos²(θ) can be rewritten using the half-angle identity: cos²(θ) = (1 + cos(2θ))/2.
Identify the angle θ in the problem, which is 5π/12. We need to find cos²(5π/12).
Apply the half-angle formula: cos²(5π/12) = (1 + cos(2 * 5π/12))/2.
Simplify the expression inside the cosine: 2 * 5π/12 = 5π/6.
Substitute back into the formula: cos²(5π/12) = (1 + cos(5π/6))/2. Now, find the value of cos(5π/6) using the unit circle or known values.
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Half-Angle Formulas
Half-angle formulas are trigonometric identities that express the sine and cosine of half an angle in terms of the sine and cosine of the original angle. For example, the cosine half-angle formula states that cos(θ/2) = ±√((1 + cos(θ))/2). These formulas are particularly useful for simplifying expressions involving angles that are not standard angles.
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Converting between Degrees & Radians
Cosine Function
The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is periodic with a period of 2π and is defined for all real numbers. Understanding the properties of the cosine function, including its values at key angles, is essential for solving trigonometric problems.
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Percorso guidato
5:53Graph of Sine and Cosine Function
Angle Conversion
Angle conversion involves changing the measure of an angle from one unit to another, such as from degrees to radians. In this context, 5π/12 radians can be converted to degrees for better understanding or verification. Knowing how to convert angles is crucial when applying trigonometric identities and formulas, especially when dealing with non-standard angles.
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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 = 1/|x|
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Domanda del libro di testo
Theory and Examples
The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.
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Domanda del libro di testo
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = 1 - cos x
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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.
cos (x − π/2) = sin x
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Domanda del libro di testo
A point P in the first quadrant lies on the parabola 𝔂 = 𝔁². Express the coordinates of P as functions of the angle of inclination of the line joining P to the origin.
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