교과서 질문
In Exercises 53–60, use a vertical shift to graph one period of the function. y = cos x + 3
766
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 55
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 552장, 문제 55
In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. tan x = -1
검증된 단계별 안내1
Recall that the equation to solve is \(\tan x = -1\) over the interval \(-2\pi \leq x \leq 2\pi\).
Understand that the tangent function has a period of \(\pi\), meaning its values repeat every \(\pi\) units along the x-axis.
Identify the reference angle where \(\tan x = 1\), which is \(\frac{\pi}{4}\), since \(\tan \frac{\pi}{4} = 1\).
Since we want \(\tan x = -1\), the solutions will be angles where the tangent is negative. Tangent is negative in the second and fourth quadrants, so the solutions correspond to angles \(x = -\frac{3\pi}{4} + k\pi\) and \(x = -\frac{7\pi}{4} + k\pi\) for integers \(k\).
Use the graph of \(y = \tan x\) to visually identify all points where the curve crosses the line \(y = -1\) within the interval \(-2\pi\) to \(2\pi\). These x-values are the solutions to the equation.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
4m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Graph of the Tangent Function
The tangent function, tan(x), is periodic with period π and has vertical asymptotes where cos(x) = 0. Its graph repeats every π units and crosses zero at multiples of π. Understanding its shape and behavior helps identify solutions to equations like tan(x) = -1 within a given interval.
추천 영상:
5:43
Introduction to Tangent Graph
Solving Trigonometric Equations Using Graphs
Solving equations graphically involves plotting the function and identifying x-values where the function equals a given value. For tan(x) = -1, you find points on the tangent curve that intersect the horizontal line y = -1 within the specified domain, providing approximate or exact solutions.
추천 영상:
04:42Solve Trig Equations Using Identity Substitutions
Domain and Periodicity in Trigonometric Solutions
The domain restriction (-2π ≤ x ≤ 2π) limits the solutions to a specific interval. Since tan(x) has period π, solutions repeat every π units. Recognizing this periodicity allows you to find all solutions within the interval by adding integer multiples of π to a principal solution.
추천 영상:
3:43Finding the Domain of an Equation
관련 실천
교과서 질문
In Exercises 53–54, let f(x) = 2 sec x, g(x) = −2 tan x, and h(x) = 2x − π/2. Graph two periods of y = (f∘h)(x).
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. cot(cot⁻¹ 9π)
733
views
교과서 질문
In Exercises 54–57, solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. A = 22.3°, c = 10
657
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
교과서 질문
In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. csc x = 1
864
views