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 105
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. 2 sin² x = 3 - sin x
검증된 단계별 안내1
Rewrite the given equation to standard form by bringing all terms to one side: \(2 \sin^{2} x + \sin x - 3 = 0\).
Recognize that this is a quadratic equation in terms of \(\sin x\). Let \(u = \sin x\), so the equation becomes \(2u^{2} + u - 3 = 0\).
Solve the quadratic equation \(2u^{2} + u - 3 = 0\) using the quadratic formula: \(u = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=2\), \(b=1\), and \(c=-3\).
Find the values of \(u\) (which represent \(\sin x\)) from the quadratic formula and determine which values are valid since \(\sin x\) must be in the interval \([-1, 1]\).
For each valid \(\sin x\) value, solve for \(x\) in the interval \([0, 2\pi)\) by using the inverse sine function and considering the sine function's symmetry to find all solutions.
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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 Trigonometric Equations
This involves rewriting the trigonometric equation in a quadratic form, such as ax² + bx + c = 0, by substituting expressions like sin²x with a variable. Once in quadratic form, standard algebraic methods like factoring or the quadratic formula can be applied to find solutions for the trigonometric function.
추천 영상:
6:24
Solving Quadratic Equations by Completing the Square
Unit Circle and Interval Restrictions
Understanding the unit circle is essential to interpret solutions for trigonometric functions within a specific interval, here [0, 2π). This helps identify all possible angles that satisfy the equation, considering the periodic nature of sine and ensuring solutions fall within the given domain.
추천 영상:
06:11Introduction to the Unit Circle
Exact Values and Approximate Solutions
Some trigonometric equations yield solutions with well-known exact values (like π/6 or π/4), while others require numerical approximation. Knowing when and how to provide exact values or approximate decimal answers (to four decimal places) is crucial for correctly presenting solutions.
추천 영상:
6:04Example 1
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. cos x - 5 = 3 cos x + 6
487
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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. sin 2x + sin x = 0
560
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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. 2 cos 2x + 1 = 0
557
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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 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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