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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Table of contents
3장, 문제 123
In Exercises 121–126, solve each equation on the interval [0, 2𝝅). 10 cos² x + 3 sin x - 9 = 0
검증된 단계별 안내1
Start by expressing the equation in terms of a single trigonometric function. Recall the Pythagorean identity: \(\cos^{2} x = 1 - \sin^{2} x\). Substitute this into the equation to rewrite it entirely in terms of \(\sin x\).
After substitution, the equation becomes \(10(1 - \sin^{2} x) + 3 \sin x - 9 = 0\). Simplify this expression by distributing and combining like terms.
Rewrite the simplified equation as a quadratic in \(\sin x\). It will take the form \(a \sin^{2} x + b \sin x + c = 0\). Identify the coefficients \(a\), \(b\), and \(c\).
Solve the quadratic equation for \(\sin x\) using the quadratic formula: \(\sin x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Calculate the discriminant and find the possible values of \(\sin x\).
Determine which solutions for \(\sin x\) lie within the valid range \([-1, 1]\). For each valid solution, find the corresponding values of \(x\) in the interval \([0, 2\pi)\) by using the inverse sine function and considering the unit circle.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, the Pythagorean identity, cos²x = 1 - sin²x, is essential to rewrite the equation in terms of a single trigonometric function, simplifying the solving process.
추천 영상:
5:32
Fundamental Trigonometric Identities
Solving Quadratic Equations in Trigonometric Form
Many trigonometric equations can be transformed into quadratic form by substituting expressions like sin x or cos x with a variable. Solving the resulting quadratic equation helps find possible values of the trigonometric function, which can then be used to determine the angle solutions.
추천 영상:
6:24Solving Quadratic Equations by Completing the Square
Interval Restriction and Solution Verification
The problem restricts solutions to the interval [0, 2π), meaning only angles within one full rotation are valid. After solving the equation, it is important to check which solutions fall within this interval and verify them to ensure they satisfy the original equation.
추천 영상:
4:34How to Solve Linear Trigonometric Equations
관련 실천
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. 3 tan² x - tan x - 2 = 0
540
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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. sec² x + 3 sec x + 2 = 0
492
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교과서 질문
In Exercises 121–126, solve each equation on the interval [0, 2𝝅). 3 cos² x - sin x = cos² x
513
views
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
524
views
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
views