교과서 질문
In Exercises 53–60, use a vertical shift to graph one period of the function. y = −3 sin 2πx + 2
832
views
Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Not the one you use?Change textbookSelect a different textbook
Table of contents
모든 교과서
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 57
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 572장, 문제 57
In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. csc x = 1
검증된 단계별 안내1
Recall that the cosecant function is the reciprocal of the sine function, so \(\csc x = \frac{1}{\sin x}\). Therefore, the equation \(\csc x = 1\) can be rewritten as \(\frac{1}{\sin x} = 1\).
From the equation \(\frac{1}{\sin x} = 1\), multiply both sides by \(\sin x\) (noting \(\sin x \neq 0\)) to get \(1 = \sin x\).
Now, solve the equation \(\sin x = 1\) for \(x\) in the interval \(-2\pi \leq x \leq 2\pi\) by identifying where the sine function reaches the value 1 on its graph.
Recall that \(\sin x = 1\) at \(x = \frac{\pi}{2} + 2k\pi\) for any integer \(k\). Find all such \(x\) values within the given interval by substituting integer values for \(k\).
List all solutions found in the interval \(-2\pi \leq x \leq 2\pi\) as the final answer to the equation \(\csc x = 1\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Understanding the Cosecant Function
The cosecant function, csc x, is the reciprocal of the sine function, defined as csc x = 1/sin x. It is undefined where sin x = 0, and its values correspond to the reciprocal of sine values. Recognizing this relationship helps in solving equations involving csc x.
추천 영상:
6:22
Graphs of Secant and Cosecant Functions
Graphing Trigonometric Functions
Graphing csc x involves plotting the reciprocal of the sine curve, which has vertical asymptotes where sine is zero. Understanding the shape and key points of the csc x graph allows one to visually identify solutions to equations like csc x = 1 within a given interval.
추천 영상:
6:04Introduction to Trigonometric Functions
Solving Trigonometric Equations on a Given Interval
Solving csc x = 1 over -2π ≤ x ≤ 2π requires finding all x-values where csc x equals 1 within this domain. This involves identifying corresponding sine values (sin x = 1) and considering the periodicity of the sine and cosecant functions to list all valid solutions.
추천 영상:
4:34How to Solve Linear Trigonometric Equations
관련 실천
교과서 질문
In Exercises 53–60, use a vertical shift to graph one period of the function. y = cos x + 3
766
views
교과서 질문
In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. cot⁻¹ (cot 3π/4)
811
views
교과서 질문
In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. tan x = -1
952
views
교과서 질문
In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. cot(cot⁻¹ 9π)
733
views
교과서 질문
In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. sec(sec⁻¹ 7π)
745
views