Solutions
Integral CalculatorDerivative CalculatorAlgebra CalculatorMatrix CalculatorMore...
Graphing
Line Graph CalculatorExponential Graph CalculatorQuadratic Graph CalculatorSin graph CalculatorMore...
Calculators
BMI CalculatorCompound Interest CalculatorPercentage CalculatorAcceleration CalculatorMore...
Geometry
Pythagorean Theorem CalculatorCircle Area CalculatorIsosceles Triangle CalculatorTriangles CalculatorMore...
Tools
NotebookGroupsCheat SheetsWorksheetsPracticeVerify
en
English
Español
Português
Français
Deutsch
Italiano
Русский
中文(简体)
한국어
日本語
Tiếng Việt
עברית
العربية
Popular Trigonometry >

sin(4x)+cos(2x)=0

  • Pre Algebra
  • Algebra
  • Pre Calculus
  • Calculus
  • Functions
  • Linear Algebra
  • Trigonometry
  • Statistics
  • Physics
  • Chemistry
  • Finance
  • Economics
  • Conversions

Solution

sin(4x)+cos(2x)=0

Solution

x=4π+4πn​,x=43π+4πn​,x=127π+12πn​,x=1211π+12πn​
+1
Degrees
x=45∘+180∘n,x=135∘+180∘n,x=105∘+180∘n,x=165∘+180∘n
Solution steps
sin(4x)+cos(2x)=0
Let: u=2xsin(2u)+cos(u)=0
Rewrite using trig identities
cos(u)+sin(2u)
Use the Double Angle identity: sin(2x)=2sin(x)cos(x)=cos(u)+2sin(u)cos(u)
cos(u)+2cos(u)sin(u)=0
Factor cos(u)+2cos(u)sin(u):cos(u)(2sin(u)+1)
cos(u)+2cos(u)sin(u)
Factor out common term cos(u)=cos(u)(1+2sin(u))
cos(u)(2sin(u)+1)=0
Solving each part separatelycos(u)=0or2sin(u)+1=0
cos(u)=0:u=2π​+2πn,u=23π​+2πn
cos(u)=0
General solutions for cos(u)=0
cos(x) periodicity table with 2πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​cos(x)123​​22​​21​0−21​−22​​−23​​​xπ67π​45π​34π​23π​35π​47π​611π​​cos(x)−1−23​​−22​​−21​021​22​​23​​​​
u=2π​+2πn,u=23π​+2πn
u=2π​+2πn,u=23π​+2πn
2sin(u)+1=0:u=67π​+2πn,u=611π​+2πn
2sin(u)+1=0
Move 1to the right side
2sin(u)+1=0
Subtract 1 from both sides2sin(u)+1−1=0−1
Simplify2sin(u)=−1
2sin(u)=−1
Divide both sides by 2
2sin(u)=−1
Divide both sides by 222sin(u)​=2−1​
Simplifysin(u)=−21​
sin(u)=−21​
General solutions for sin(u)=−21​
sin(x) periodicity table with 2πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​sin(x)021​22​​23​​123​​22​​21​​xπ67π​45π​34π​23π​35π​47π​611π​​sin(x)0−21​−22​​−23​​−1−23​​−22​​−21​​​
u=67π​+2πn,u=611π​+2πn
u=67π​+2πn,u=611π​+2πn
Combine all the solutionsu=2π​+2πn,u=23π​+2πn,u=67π​+2πn,u=611π​+2πn
Substitute back u=2x
2x=2π​+2πn:x=4π+4πn​
2x=2π​+2πn
Divide both sides by 2
2x=2π​+2πn
Divide both sides by 222x​=22π​​+22πn​
Simplify
22x​=22π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 22π​​+22πn​:4π+4πn​
22π​​+22πn​
Apply rule ca​±cb​=ca±b​=22π​+2πn​
Join 2π​+2πn:2π+4πn​
2π​+2πn
Convert element to fraction: 2πn=22πn2​=2π​+22πn⋅2​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=2π+2πn⋅2​
Multiply the numbers: 2⋅2=4=2π+4πn​
=22π+4πn​​
Apply the fraction rule: acb​​=c⋅ab​=2⋅2π+4πn​
Multiply the numbers: 2⋅2=4=4π+4πn​
x=4π+4πn​
x=4π+4πn​
x=4π+4πn​
2x=23π​+2πn:x=43π+4πn​
2x=23π​+2πn
Divide both sides by 2
2x=23π​+2πn
Divide both sides by 222x​=223π​​+22πn​
Simplify
22x​=223π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 223π​​+22πn​:43π+4πn​
223π​​+22πn​
Apply rule ca​±cb​=ca±b​=223π​+2πn​
Join 23π​+2πn:23π+4πn​
23π​+2πn
Convert element to fraction: 2πn=22πn2​=23π​+22πn⋅2​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=23π+2πn⋅2​
Multiply the numbers: 2⋅2=4=23π+4πn​
=223π+4πn​​
Apply the fraction rule: acb​​=c⋅ab​=2⋅23π+4πn​
Multiply the numbers: 2⋅2=4=43π+4πn​
x=43π+4πn​
x=43π+4πn​
x=43π+4πn​
2x=67π​+2πn:x=127π+12πn​
2x=67π​+2πn
Divide both sides by 2
2x=67π​+2πn
Divide both sides by 222x​=267π​​+22πn​
Simplify
22x​=267π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 267π​​+22πn​:127π+12πn​
267π​​+22πn​
Apply rule ca​±cb​=ca±b​=267π​+2πn​
Join 67π​+2πn:67π+12πn​
67π​+2πn
Convert element to fraction: 2πn=62πn6​=67π​+62πn⋅6​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=67π+2πn⋅6​
Multiply the numbers: 2⋅6=12=67π+12πn​
=267π+12πn​​
Apply the fraction rule: acb​​=c⋅ab​=6⋅27π+12πn​
Multiply the numbers: 6⋅2=12=127π+12πn​
x=127π+12πn​
x=127π+12πn​
x=127π+12πn​
2x=611π​+2πn:x=1211π+12πn​
2x=611π​+2πn
Divide both sides by 2
2x=611π​+2πn
Divide both sides by 222x​=2611π​​+22πn​
Simplify
22x​=2611π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 2611π​​+22πn​:1211π+12πn​
2611π​​+22πn​
Apply rule ca​±cb​=ca±b​=2611π​+2πn​
Join 611π​+2πn:611π+12πn​
611π​+2πn
Convert element to fraction: 2πn=62πn6​=611π​+62πn⋅6​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=611π+2πn⋅6​
Multiply the numbers: 2⋅6=12=611π+12πn​
=2611π+12πn​​
Apply the fraction rule: acb​​=c⋅ab​=6⋅211π+12πn​
Multiply the numbers: 6⋅2=12=1211π+12πn​
x=1211π+12πn​
x=1211π+12πn​
x=1211π+12πn​
x=4π+4πn​,x=43π+4πn​,x=127π+12πn​,x=1211π+12πn​

Graph

Sorry, your browser does not support this application
View interactive graph

Popular Examples

sin^2(θ)-11sin(θ)=0sin(θ)=msin(θ)=(sqrt(3))/2 ,cos(θ)<0,0<θ<2pitan(x)=1,0<= x<2pisolvefor x,sin(2x)*cos(2x)=0.5
Study ToolsAI Math SolverPopular ProblemsWorksheetsStudy GuidesPracticeCheat SheetsCalculatorsGraphing CalculatorGeometry CalculatorVerify Solution
AppsSymbolab App (Android)Graphing Calculator (Android)Practice (Android)Symbolab App (iOS)Graphing Calculator (iOS)Practice (iOS)Chrome ExtensionSymbolab Math Solver API
CompanyAbout SymbolabBlogHelp
LegalPrivacyTermsCookie PolicyCookie SettingsDo Not Sell or Share My Personal InfoCopyright, Community Guidelines, DSA & other Legal ResourcesLearneo Legal Center
Social Media
Symbolab, a Learneo, Inc. business
© Learneo, Inc. 2024