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1. Find all solutions to the following trigonometric equations. (a) tan? r – 6 tan r + 5 = 0 (b) 2 sinr + 5 cos r - 4 = 0

1. Find all solutions to the following trigonometric equations. (a) tan? r – 6 tan r + 5 = 0 (b) 2 sinr + 5 cos r - 4 = 0

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
# Solving Trigonometric Equations ## Problem Statement Find all solutions to the following trigonometric equations: ### (a) \[ \tan^2 x - 6 \tan x + 5 = 0 \] ### (b) \[ 2 \sin^2 x + 5 \cos x - 4 = 0 \] ### Explanation The task involves solving trigonometric equations, which may require methods such as factoring, using trigonometric identities, or applying the quadratic formula. Each equation is set to zero, indicating that you need to find values of \( x \) where each equation holds true. #### For Equation (a): - The equation is quadratic in terms of \( \tan x \). - It can be approached by substituting \( u = \tan x \), turning it into a quadratic equation: \[ u^2 - 6u + 5 = 0 \] - Solve for \( u \) to find possible values of \( \tan x \). #### For Equation (b): - The equation involves both \( \sin^2 x \) and \( \cos x \). - Notice that \( \sin^2 x = 1 - \cos^2 x \). Use this identity to rewrite the equation if necessary: \[ 2(1 - \cos^2 x) + 5 \cos x - 4 = 0 \] - After substitution, simplify and solve the resulting equation for \( \cos x \) and consequently for \( x \). ### Note No graphs or diagrams are provided in the image. All calculations and solutions should follow trigonometric solving techniques and consider the periodicity of trigonometric functions when finding the general solution.
Transcribed Image Text:# Solving Trigonometric Equations ## Problem Statement Find all solutions to the following trigonometric equations: ### (a) \[ \tan^2 x - 6 \tan x + 5 = 0 \] ### (b) \[ 2 \sin^2 x + 5 \cos x - 4 = 0 \] ### Explanation The task involves solving trigonometric equations, which may require methods such as factoring, using trigonometric identities, or applying the quadratic formula. Each equation is set to zero, indicating that you need to find values of \( x \) where each equation holds true. #### For Equation (a): - The equation is quadratic in terms of \( \tan x \). - It can be approached by substituting \( u = \tan x \), turning it into a quadratic equation: \[ u^2 - 6u + 5 = 0 \] - Solve for \( u \) to find possible values of \( \tan x \). #### For Equation (b): - The equation involves both \( \sin^2 x \) and \( \cos x \). - Notice that \( \sin^2 x = 1 - \cos^2 x \). Use this identity to rewrite the equation if necessary: \[ 2(1 - \cos^2 x) + 5 \cos x - 4 = 0 \] - After substitution, simplify and solve the resulting equation for \( \cos x \) and consequently for \( x \). ### Note No graphs or diagrams are provided in the image. All calculations and solutions should follow trigonometric solving techniques and consider the periodicity of trigonometric functions when finding the general solution.
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