교과서 질문
The Greatest and Least Integer Functions
Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.
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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
1장, 문제 1.3.54
Solving Trigonometric Equations
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
sin² θ = cos² θ
검증된 단계별 안내1
Start by recognizing the given equation: sin² θ = cos² θ. This can be rewritten using the identity sin² θ + cos² θ = 1.
Rearrange the identity to express one trigonometric function in terms of the other: sin² θ = 1 - cos² θ.
Substitute this expression into the original equation: 1 - cos² θ = cos² θ.
Combine like terms to form a single equation: 1 = 2cos² θ.
Solve for cos² θ by dividing both sides by 2, then take the square root to find the possible values of cos θ. Consider the range 0 ≤ θ ≤ 2π to determine the specific angles that satisfy the equation.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
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Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. A key identity relevant to the given equation is the Pythagorean identity, which states that sin² θ + cos² θ = 1. Understanding these identities is crucial for simplifying and solving trigonometric equations.
추천 영상:
7:17
Verifying Trig Equations as Identities
Solving Trigonometric Equations
Solving trigonometric equations involves finding the angles that satisfy the equation within a specified interval. In this case, we need to manipulate the equation sin² θ = cos² θ to find the values of θ between 0 and 2π. Techniques often include using identities, factoring, and applying inverse trigonometric functions.
추천 영상:
5:02Solving Logarithmic Equations
Unit Circle
The unit circle is a fundamental concept in trigonometry that provides a geometric interpretation of the sine and cosine functions. It is a circle with a radius of one centered at the origin of a coordinate plane. Understanding the unit circle helps in determining the angles corresponding to specific sine and cosine values, which is essential for solving the equation sin² θ = cos² θ.
추천 영상:
5:10Evaluate Composite Functions - Values on Unit Circle
관련 실천
교과서 질문
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = sec x tan x
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교과서 질문
Functions and Graphs
Find the natural domain and graph the functions in Exercises 15–20.
f(x) = 1 − 2x − x²
386
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교과서 질문
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
g(x) = x⁴ + 3x² − 1
298
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교과서 질문
Graph the function y = √|x|.
280
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교과서 질문
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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