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 >

cos(2x)+cos(4x)+cos(6x)=0

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

Solution

cos(2x)+cos(4x)+cos(6x)=0

Solution

x=3π(3n+1)​,x=3π(3n+2)​,x=83π+8πn​,x=85π+8πn​,x=8π+8πn​,x=87π+8πn​
+1
Degrees
x=60∘+180∘n,x=120∘+180∘n,x=67.5∘+180∘n,x=112.5∘+180∘n,x=22.5∘+180∘n,x=157.5∘+180∘n
Solution steps
cos(2x)+cos(4x)+cos(6x)=0
Let: u=2xcos(u)+cos(2u)+cos(3u)=0
Rewrite using trig identities
cos(2u)+cos(3u)+cos(u)
Use the Double Angle identity: cos(2x)=2cos2(x)−1=2cos2(u)−1+cos(3u)+cos(u)
cos(3u)=4cos3(u)−3cos(u)
cos(3u)
Rewrite using trig identities
cos(3u)
Rewrite as=cos(2u+u)
Use the Angle Sum identity: cos(s+t)=cos(s)cos(t)−sin(s)sin(t)=cos(2u)cos(u)−sin(2u)sin(u)
Use the Double Angle identity: sin(2u)=2sin(u)cos(u)=cos(2u)cos(u)−2sin(u)cos(u)sin(u)
Simplify cos(2u)cos(u)−2sin(u)cos(u)sin(u):cos(u)cos(2u)−2sin2(u)cos(u)
cos(2u)cos(u)−2sin(u)cos(u)sin(u)
2sin(u)cos(u)sin(u)=2sin2(u)cos(u)
2sin(u)cos(u)sin(u)
Apply exponent rule: ab⋅ac=ab+csin(u)sin(u)=sin1+1(u)=2cos(u)sin1+1(u)
Add the numbers: 1+1=2=2cos(u)sin2(u)
=cos(u)cos(2u)−2sin2(u)cos(u)
=cos(u)cos(2u)−2sin2(u)cos(u)
=cos(u)cos(2u)−2sin2(u)cos(u)
Use the Double Angle identity: cos(2u)=2cos2(u)−1=(2cos2(u)−1)cos(u)−2sin2(u)cos(u)
Use the Pythagorean identity: cos2(u)+sin2(u)=1sin2(u)=1−cos2(u)=(2cos2(u)−1)cos(u)−2(1−cos2(u))cos(u)
Expand (2cos2(u)−1)cos(u)−2(1−cos2(u))cos(u):4cos3(u)−3cos(u)
(2cos2(u)−1)cos(u)−2(1−cos2(u))cos(u)
=cos(u)(2cos2(u)−1)−2cos(u)(1−cos2(u))
Expand cos(u)(2cos2(u)−1):2cos3(u)−cos(u)
cos(u)(2cos2(u)−1)
Apply the distributive law: a(b−c)=ab−aca=cos(u),b=2cos2(u),c=1=cos(u)2cos2(u)−cos(u)1
=2cos2(u)cos(u)−1cos(u)
Simplify 2cos2(u)cos(u)−1⋅cos(u):2cos3(u)−cos(u)
2cos2(u)cos(u)−1cos(u)
2cos2(u)cos(u)=2cos3(u)
2cos2(u)cos(u)
Apply exponent rule: ab⋅ac=ab+ccos2(u)cos(u)=cos2+1(u)=2cos2+1(u)
Add the numbers: 2+1=3=2cos3(u)
1⋅cos(u)=cos(u)
1cos(u)
Multiply: 1⋅cos(u)=cos(u)=cos(u)
=2cos3(u)−cos(u)
=2cos3(u)−cos(u)
=2cos3(u)−cos(u)−2(1−cos2(u))cos(u)
Expand −2cos(u)(1−cos2(u)):−2cos(u)+2cos3(u)
−2cos(u)(1−cos2(u))
Apply the distributive law: a(b−c)=ab−aca=−2cos(u),b=1,c=cos2(u)=−2cos(u)1−(−2cos(u))cos2(u)
Apply minus-plus rules−(−a)=a=−2⋅1cos(u)+2cos2(u)cos(u)
Simplify −2⋅1⋅cos(u)+2cos2(u)cos(u):−2cos(u)+2cos3(u)
−2⋅1cos(u)+2cos2(u)cos(u)
2⋅1⋅cos(u)=2cos(u)
2⋅1cos(u)
Multiply the numbers: 2⋅1=2=2cos(u)
2cos2(u)cos(u)=2cos3(u)
2cos2(u)cos(u)
Apply exponent rule: ab⋅ac=ab+ccos2(u)cos(u)=cos2+1(u)=2cos2+1(u)
Add the numbers: 2+1=3=2cos3(u)
=−2cos(u)+2cos3(u)
=−2cos(u)+2cos3(u)
=2cos3(u)−cos(u)−2cos(u)+2cos3(u)
Simplify 2cos3(u)−cos(u)−2cos(u)+2cos3(u):4cos3(u)−3cos(u)
2cos3(u)−cos(u)−2cos(u)+2cos3(u)
Group like terms=2cos3(u)+2cos3(u)−cos(u)−2cos(u)
Add similar elements: 2cos3(u)+2cos3(u)=4cos3(u)=4cos3(u)−cos(u)−2cos(u)
Add similar elements: −cos(u)−2cos(u)=−3cos(u)=4cos3(u)−3cos(u)
=4cos3(u)−3cos(u)
=4cos3(u)−3cos(u)
=−1+4cos3(u)−3cos(u)+cos(u)+2cos2(u)
Simplify=−1+4cos3(u)−2cos(u)+2cos2(u)
−1−2cos(u)+2cos2(u)+4cos3(u)=0
Solve by substitution
−1−2cos(u)+2cos2(u)+4cos3(u)=0
Let: cos(u)=u−1−2u+2u2+4u3=0
−1−2u+2u2+4u3=0:u=−21​,u=−22​​,u=22​​
−1−2u+2u2+4u3=0
Write in the standard form an​xn+…+a1​x+a0​=04u3+2u2−2u−1=0
Factor 4u3+2u2−2u−1:(2u+1)(2​u+1)(2​u−1)
4u3+2u2−2u−1
=(4u3+2u2)+(−2u−1)
Factor out −1from −2u−1:−(2u+1)
−2u−1
Factor out common term −1=−(2u+1)
Factor out 2u2from 4u3+2u2:2u2(2u+1)
4u3+2u2
Apply exponent rule: ab+c=abacu3=uu2=4uu2+2u2
Rewrite 4 as 2⋅2=2⋅2uu2+2u2
Factor out common term 2u2=2u2(2u+1)
=−(2u+1)+2u2(2u+1)
Factor out common term 2u+1=(2u+1)(2u2−1)
Factor 2u2−1:(2​u+1)(2​u−1)
2u2−1
Rewrite 2u2−1 as (2​u)2−12
2u2−1
Apply radical rule: a=(a​)22=(2​)2=(2​)2u2−1
Rewrite 1 as 12=(2​)2u2−12
Apply exponent rule: ambm=(ab)m(2​)2u2=(2​u)2=(2​u)2−12
=(2​u)2−12
Apply Difference of Two Squares Formula: x2−y2=(x+y)(x−y)(2​u)2−12=(2​u+1)(2​u−1)=(2​u+1)(2​u−1)
=(2u+1)(2​u+1)(2​u−1)
(2u+1)(2​u+1)(2​u−1)=0
Using the Zero Factor Principle: If ab=0then a=0or b=02u+1=0or2​u+1=0or2​u−1=0
Solve 2u+1=0:u=−21​
2u+1=0
Move 1to the right side
2u+1=0
Subtract 1 from both sides2u+1−1=0−1
Simplify2u=−1
2u=−1
Divide both sides by 2
2u=−1
Divide both sides by 222u​=2−1​
Simplifyu=−21​
u=−21​
Solve 2​u+1=0:u=−22​​
2​u+1=0
Move 1to the right side
2​u+1=0
Subtract 1 from both sides2​u+1−1=0−1
Simplify2​u=−1
2​u=−1
Divide both sides by 2​
2​u=−1
Divide both sides by 2​2​2​u​=2​−1​
Simplify
2​2​u​=2​−1​
Simplify 2​2​u​:u
2​2​u​
Cancel the common factor: 2​=u
Simplify 2​−1​:−22​​
2​−1​
Apply the fraction rule: b−a​=−ba​=−2​1​
Rationalize −2​1​:−22​​
−2​1​
Multiply by the conjugate 2​2​​=−2​2​1⋅2​​
1⋅2​=2​
2​2​=2
2​2​
Apply radical rule: a​a​=a2​2​=2=2
=−22​​
=−22​​
u=−22​​
u=−22​​
u=−22​​
Solve 2​u−1=0:u=22​​
2​u−1=0
Move 1to the right side
2​u−1=0
Add 1 to both sides2​u−1+1=0+1
Simplify2​u=1
2​u=1
Divide both sides by 2​
2​u=1
Divide both sides by 2​2​2​u​=2​1​
Simplify
2​2​u​=2​1​
Simplify 2​2​u​:u
2​2​u​
Cancel the common factor: 2​=u
Simplify 2​1​:22​​
2​1​
Multiply by the conjugate 2​2​​=2​2​1⋅2​​
1⋅2​=2​
2​2​=2
2​2​
Apply radical rule: a​a​=a2​2​=2=2
=22​​
u=22​​
u=22​​
u=22​​
The solutions areu=−21​,u=−22​​,u=22​​
Substitute back u=cos(u)cos(u)=−21​,cos(u)=−22​​,cos(u)=22​​
cos(u)=−21​,cos(u)=−22​​,cos(u)=22​​
cos(u)=−21​:u=32π​+2πn,u=34π​+2πn
cos(u)=−21​
General solutions for cos(u)=−21​
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=32π​+2πn,u=34π​+2πn
u=32π​+2πn,u=34π​+2πn
cos(u)=−22​​:u=43π​+2πn,u=45π​+2πn
cos(u)=−22​​
General solutions for cos(u)=−22​​
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=43π​+2πn,u=45π​+2πn
u=43π​+2πn,u=45π​+2πn
cos(u)=22​​:u=4π​+2πn,u=47π​+2πn
cos(u)=22​​
General solutions for cos(u)=22​​
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=4π​+2πn,u=47π​+2πn
u=4π​+2πn,u=47π​+2πn
Combine all the solutionsu=32π​+2πn,u=34π​+2πn,u=43π​+2πn,u=45π​+2πn,u=4π​+2πn,u=47π​+2πn
Substitute back u=2x
2x=32π​+2πn:x=3π(3n+1)​
2x=32π​+2πn
Divide both sides by 2
2x=32π​+2πn
Divide both sides by 222x​=232π​​+22πn​
Simplify
22x​=232π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 232π​​+22πn​:3π(3n+1)​
232π​​+22πn​
Apply rule ca​±cb​=ca±b​=232π​+2πn​
Join 32π​+2πn:32π+6πn​
32π​+2πn
Convert element to fraction: 2πn=32πn3​=32π​+32πn⋅3​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=32π+2πn⋅3​
Multiply the numbers: 2⋅3=6=32π+6πn​
=232π+6πn​​
Apply the fraction rule: acb​​=c⋅ab​=3⋅22π+6πn​
Multiply the numbers: 3⋅2=6=62π+6πn​
Factor 2π+6πn:2π(1+3n)
2π+6πn
Rewrite as=1⋅2π+3⋅2πn
Factor out common term 2π=2π(1+3n)
=62π(1+3n)​
Cancel the common factor: 2=3π(3n+1)​
x=3π(3n+1)​
x=3π(3n+1)​
x=3π(3n+1)​
2x=34π​+2πn:x=3π(3n+2)​
2x=34π​+2πn
Divide both sides by 2
2x=34π​+2πn
Divide both sides by 222x​=234π​​+22πn​
Simplify
22x​=234π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 234π​​+22πn​:3π(3n+2)​
234π​​+22πn​
Apply rule ca​±cb​=ca±b​=234π​+2πn​
Join 34π​+2πn:34π+6πn​
34π​+2πn
Convert element to fraction: 2πn=32πn3​=34π​+32πn⋅3​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=34π+2πn⋅3​
Multiply the numbers: 2⋅3=6=34π+6πn​
=234π+6πn​​
Apply the fraction rule: acb​​=c⋅ab​=3⋅24π+6πn​
Multiply the numbers: 3⋅2=6=64π+6πn​
Factor 4π+6πn:2π(2+3n)
4π+6πn
Rewrite as=2⋅2π+3⋅2πn
Factor out common term 2π=2π(2+3n)
=62π(2+3n)​
Cancel the common factor: 2=3π(3n+2)​
x=3π(3n+2)​
x=3π(3n+2)​
x=3π(3n+2)​
2x=43π​+2πn:x=83π+8πn​
2x=43π​+2πn
Divide both sides by 2
2x=43π​+2πn
Divide both sides by 222x​=243π​​+22πn​
Simplify
22x​=243π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 243π​​+22πn​:83π+8πn​
243π​​+22πn​
Apply rule ca​±cb​=ca±b​=243π​+2πn​
Join 43π​+2πn:43π+8πn​
43π​+2πn
Convert element to fraction: 2πn=42πn4​=43π​+42πn⋅4​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=43π+2πn⋅4​
Multiply the numbers: 2⋅4=8=43π+8πn​
=243π+8πn​​
Apply the fraction rule: acb​​=c⋅ab​=4⋅23π+8πn​
Multiply the numbers: 4⋅2=8=83π+8πn​
x=83π+8πn​
x=83π+8πn​
x=83π+8πn​
2x=45π​+2πn:x=85π+8πn​
2x=45π​+2πn
Divide both sides by 2
2x=45π​+2πn
Divide both sides by 222x​=245π​​+22πn​
Simplify
22x​=245π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 245π​​+22πn​:85π+8πn​
245π​​+22πn​
Apply rule ca​±cb​=ca±b​=245π​+2πn​
Join 45π​+2πn:45π+8πn​
45π​+2πn
Convert element to fraction: 2πn=42πn4​=45π​+42πn⋅4​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=45π+2πn⋅4​
Multiply the numbers: 2⋅4=8=45π+8πn​
=245π+8πn​​
Apply the fraction rule: acb​​=c⋅ab​=4⋅25π+8πn​
Multiply the numbers: 4⋅2=8=85π+8πn​
x=85π+8πn​
x=85π+8πn​
x=85π+8πn​
2x=4π​+2πn:x=8π+8πn​
2x=4π​+2πn
Divide both sides by 2
2x=4π​+2πn
Divide both sides by 222x​=24π​​+22πn​
Simplify
22x​=24π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 24π​​+22πn​:8π+8πn​
24π​​+22πn​
Apply rule ca​±cb​=ca±b​=24π​+2πn​
Join 4π​+2πn:4π+8πn​
4π​+2πn
Convert element to fraction: 2πn=42πn4​=4π​+42πn⋅4​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=4π+2πn⋅4​
Multiply the numbers: 2⋅4=8=4π+8πn​
=24π+8πn​​
Apply the fraction rule: acb​​=c⋅ab​=4⋅2π+8πn​
Multiply the numbers: 4⋅2=8=8π+8πn​
x=8π+8πn​
x=8π+8πn​
x=8π+8πn​
2x=47π​+2πn:x=87π+8πn​
2x=47π​+2πn
Divide both sides by 2
2x=47π​+2πn
Divide both sides by 222x​=247π​​+22πn​
Simplify
22x​=247π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 247π​​+22πn​:87π+8πn​
247π​​+22πn​
Apply rule ca​±cb​=ca±b​=247π​+2πn​
Join 47π​+2πn:47π+8πn​
47π​+2πn
Convert element to fraction: 2πn=42πn4​=47π​+42πn⋅4​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=47π+2πn⋅4​
Multiply the numbers: 2⋅4=8=47π+8πn​
=247π+8πn​​
Apply the fraction rule: acb​​=c⋅ab​=4⋅27π+8πn​
Multiply the numbers: 4⋅2=8=87π+8πn​
x=87π+8πn​
x=87π+8πn​
x=87π+8πn​
x=3π(3n+1)​,x=3π(3n+2)​,x=83π+8πn​,x=85π+8πn​,x=8π+8πn​,x=87π+8πn​

Graph

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

Popular Examples

sqrt(2)cos(x)sin(x)-cos(x)=06sin(x)=6sin(2x)sin(5x)+sin(x)=sqrt(3)cos(2x)3sin(2x)-1=0-sec(x)=csc(3.45)
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