교과서 질문
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 4 tan² x - 8 tan x + 3 = 0
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
3장, 문제 91
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). cos² x - cos x - 1 = 0
검증된 단계별 안내1
Recognize that the equation is a quadratic in terms of \( \cos x \). Let \( y = \cos x \), so the equation becomes \( y^2 - y - 1 = 0 \).
Use the quadratic formula to solve for \( y \): \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -1 \), and \( c = -1 \).
Calculate the discriminant \( \Delta = b^2 - 4ac = (-1)^2 - 4(1)(-1) = 1 + 4 = 5 \), then find the two roots \( y_1 \) and \( y_2 \).
Since \( y = \cos x \), check which roots are within the valid range for cosine values, i.e., between -1 and 1. Discard any root outside this range.
For each valid root \( y \), solve \( \cos x = y \) on the interval \( [0, 2\pi) \) using the inverse cosine function and symmetry properties of cosine, then use a calculator to find the approximate values of \( x \) correct to four decimal places.
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주요 개념
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Trigonometric Equations
Trigonometric equations involve functions like sine, cosine, and tangent. Solving these equations means finding all angle values within a specified interval that satisfy the equation. Understanding how to manipulate and rearrange these equations is essential for isolating the trigonometric function.
추천 영상:
4:34
How to Solve Linear Trigonometric Equations
Quadratic Form in Trigonometry
Some trigonometric equations can be rewritten as quadratic equations by substituting a trigonometric function (e.g., cos x) with a variable. This allows the use of algebraic methods like factoring or the quadratic formula to find solutions for the trigonometric function before solving for the angle.
추천 영상:
6:36Quadratic Formula
Using a Calculator and Interval Restrictions
After finding the trigonometric function values, a calculator is used to determine the corresponding angles, ensuring answers are accurate to four decimal places. Solutions must be restricted to the given interval [0, 2π), meaning only angles within one full rotation are considered.
추천 영상:
4:45How to Use a Calculator for Trig Functions
관련 실천
교과서 질문
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). sin x = 0.8246
458
views
교과서 질문
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 7 sin² x - 1 = 0
497
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교과서 질문
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). cos x = ﹣ 2/5
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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 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). tan x = ﹣3
472
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