Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = 2e⁻ˣ - 3
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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 1.3.42
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
cos (3π/2 + x)
Verified step by step guidance1
Recognize that the expression cos(3π/2 + x) involves a trigonometric identity for cosine of a sum, which is cos(a + b) = cos(a)cos(b) - sin(a)sin(b).
Identify the values of a and b in the expression: here, a = 3π/2 and b = x.
Substitute these values into the identity: cos(3π/2 + x) = cos(3π/2)cos(x) - sin(3π/2)sin(x).
Recall the exact trigonometric values: cos(3π/2) = 0 and sin(3π/2) = -1.
Substitute these values back into the expression: cos(3π/2 + x) = 0 * cos(x) - (-1) * sin(x), which simplifies to 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. They are essential for simplifying expressions and solving equations in trigonometry. Key identities include the Pythagorean identity, angle sum and difference identities, and double angle formulas, which help express trigonometric functions in various forms.
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Angle Addition Formulas
Angle addition formulas are specific trigonometric identities that express the sine and cosine of the sum or difference of two angles. For example, the cosine of the sum of two angles is given by cos(a + b) = cos(a)cos(b) - sin(a)sin(b). These formulas are crucial for transforming expressions involving sums of angles into simpler forms using basic trigonometric functions.
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Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it provides a geometric interpretation of the sine and cosine functions. The coordinates of points on the unit circle correspond to the values of cosine and sine for various angles, making it easier to evaluate trigonometric functions for any angle.
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Related Practice
Textbook Question
Functions
In Exercises 1–6, find the domain and range of each function.
f(t) = 4/(3 − t)
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Textbook Question
Algebraic Combinations
In Exercises 3 and 4, find the domains of f, g, f/g and g/f.
f(x) = 1, g(x) = 1 + √x
Textbook Question
Finding Formulas for Functions
Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.
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Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x cos x
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Textbook Question
Finding Formulas for Functions
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.
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