Textbook Question
Functions
In Exercises 1–6, find the domain and range of each function.
f(x) = 1 + x²
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.3.40
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
sin (2π − x)
Verified step by step guidance1
Recognize that the expression involves the sine of a difference: sin(2π - x). This can be simplified using the sine subtraction formula: sin(a - b) = sin(a)cos(b) - cos(a)sin(b).
Substitute a = 2π and b = x into the formula: sin(2π - x) = sin(2π)cos(x) - cos(2π)sin(x).
Recall the values of sin(2π) and cos(2π). Since 2π is a full rotation in the unit circle, sin(2π) = 0 and cos(2π) = 1.
Substitute these values into the expression: sin(2π - x) = 0 * cos(x) - 1 * sin(x).
Simplify the expression: sin(2π - x) = -sin(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One important identity is the sine and cosine of complementary angles, which states that sin(π - x) = sin(x) and cos(π - x) = -cos(x). These identities are essential for simplifying expressions involving trigonometric functions.
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Verifying Trig Equations as Identities
Sine Function Properties
The sine function, denoted as sin(x), is a periodic function that represents the y-coordinate of a point on the unit circle corresponding to an angle x. It has specific properties, such as sin(2π - x) = sin(x), which reflects the symmetry of the sine function about the y-axis. Understanding these properties helps in transforming trigonometric expressions.
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Angle Subtraction
Angle subtraction refers to the process of finding the sine or cosine of an angle expressed as the difference between two angles. In this case, sin(2π - x) can be analyzed using the periodic nature of the sine function, which allows us to express it in terms of sin(x) and cos(x). Recognizing how angles relate to one another is crucial for simplifying trigonometric expressions.
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Related Practice
Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x² + 1
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Textbook Question
Functions and Graphs
Express the area and circumference of a circle as functions of the circle’s radius. Then express the area as a function of the circumference.
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Textbook Question
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 = L/2π sin (2πt/L), L > 0
Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x⁴ + 1
x³ - 2x
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Textbook Question
Graph the functions in Exercises 37–56.
y = (x + 2)³/² + 1
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