교과서 질문
In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 7 cos x = 4 - 2 sin² x
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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3장, 문제 116
In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 3 tan² x - tan x - 2 = 0
검증된 단계별 안내1
Recognize that the given equation is a quadratic in terms of \( \tan x \): \( 3 \tan^{2} x - \tan x - 2 = 0 \). Our goal is to solve for \( x \) in the interval \( [0, 2\pi) \).
Let \( t = \tan x \). Rewrite the equation as \( 3t^{2} - t - 2 = 0 \). This is a standard quadratic equation in \( t \).
Use the quadratic formula to solve for \( t \): \[ t = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \( a = 3 \), \( b = -1 \), and \( c = -2 \).
Calculate the discriminant \( \Delta = b^{2} - 4ac \) and find the two possible values for \( t = \tan x \).
For each value of \( t \), solve \( \tan x = t \) on the interval \( [0, 2\pi) \). Recall that \( \tan x \) has period \( \pi \), so the solutions are \( x = \arctan(t) + k\pi \) for integers \( k \). Find the specific solutions within the given interval.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Solving Quadratic Equations in Trigonometric Functions
This concept involves treating trigonometric equations like algebraic quadratics by substituting the trigonometric function (e.g., tan x) as a variable. The equation 3 tan² x - tan x - 2 = 0 can be solved using factoring or the quadratic formula to find values of tan x, which then lead to solutions for x.
추천 영상:
6:24
Solving Quadratic Equations by Completing the Square
Properties and Periodicity of the Tangent Function
Understanding that the tangent function has a period of π and is undefined at odd multiples of π/2 is crucial. Solutions for x must be found within the interval [0, 2π), considering the periodicity to identify all valid angles where tan x satisfies the equation.
추천 영상:
5:43Introduction to Tangent Graph
Finding Exact and Approximate Solutions
After solving for tan x, one must find the corresponding angles x using inverse tangent functions. Exact values are preferred when possible, but approximate solutions to four decimal places are acceptable, especially when the inverse tangent yields irrational numbers.
추천 영상:
05:13Finding Direction of a Vector
관련 실천
교과서 질문
In Exercises 121–126, solve each equation on the interval [0, 2𝝅). 3 cos² x - sin x = cos² x
513
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교과서 질문
In Exercises 127–130, solve each equation on the interval [0, 2𝝅) by first rewriting the equation in terms of sines or cosines. csc² x + csc x - 2 = 0
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교과서 질문
In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. tan x sec x = 2 tan x
536
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교과서 질문
In Exercises 121–126, solve each equation on the interval [0, 2𝝅). 10 cos² x + 3 sin x - 9 = 0
529
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교과서 질문
In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 5 cot² x - 15 = 0
449
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