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 >

0=1.9cos(0.1t)-1.25sin(0.2t)

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

Solution

0=1.9cos(0.1t)−1.25sin(0.2t)

Solution

t=5π+20πn,t=15π+20πn,t=0.10.86331…+2πn​,t=0.1π−0.86331…+2πn​
+1
Degrees
t=900∘+3600∘n,t=2700∘+3600∘n,t=494.64197…∘+3600∘n,t=1305.35802…∘+3600∘n
Solution steps
0=1.9cos(0.1t)−1.25sin(0.2t)
Switch sides1.9cos(0.1t)−1.25sin(0.2t)=0
Let: u=0.1t1.9cos(u)−1.25sin(2u)=0
Rewrite using trig identities
−1.25sin(2u)+1.9cos(u)
Use the Double Angle identity: sin(2x)=2sin(x)cos(x)=−1.25⋅2sin(u)cos(u)+1.9cos(u)
Simplify=−2.5sin(u)cos(u)+1.9cos(u)
1.9cos(u)−2.5cos(u)sin(u)=0
Factor 1.9cos(u)−2.5cos(u)sin(u):−cos(u)(2.5sin(u)−1.9)
1.9cos(u)−2.5cos(u)sin(u)
Factor out common term −cos(u)=−cos(u)(−1.9+2.5sin(u))
−cos(u)(2.5sin(u)−1.9)=0
Solving each part separatelycos(u)=0or2.5sin(u)−1.9=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
2.5sin(u)−1.9=0:u=arcsin(2519​)+2πn,u=π−arcsin(2519​)+2πn
2.5sin(u)−1.9=0
Multiply both sides by 10
2.5sin(u)−1.9=0
To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by 102.5sin(u)⋅10−1.9⋅10=0⋅10
Refine25sin(u)−19=0
25sin(u)−19=0
Move 19to the right side
25sin(u)−19=0
Add 19 to both sides25sin(u)−19+19=0+19
Simplify25sin(u)=19
25sin(u)=19
Divide both sides by 25
25sin(u)=19
Divide both sides by 252525sin(u)​=2519​
Simplifysin(u)=2519​
sin(u)=2519​
Apply trig inverse properties
sin(u)=2519​
General solutions for sin(u)=2519​sin(x)=a⇒x=arcsin(a)+2πn,x=π−arcsin(a)+2πnu=arcsin(2519​)+2πn,u=π−arcsin(2519​)+2πn
u=arcsin(2519​)+2πn,u=π−arcsin(2519​)+2πn
Combine all the solutionsu=2π​+2πn,u=23π​+2πn,u=arcsin(2519​)+2πn,u=π−arcsin(2519​)+2πn
Substitute back u=0.1t
0.1t=2π​+2πn:t=5π+20πn
0.1t=2π​+2πn
Multiply both sides by 10
0.1t=2π​+2πn
To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by 100.1t⋅10=2π​⋅10+2πn⋅10
Refinet=5π+20πn
t=5π+20πn
0.1t=23π​+2πn:t=15π+20πn
0.1t=23π​+2πn
Multiply both sides by 10
0.1t=23π​+2πn
To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by 100.1t⋅10=23π​⋅10+2πn⋅10
Refinet=15π+20πn
t=15π+20πn
0.1t=arcsin(2519​)+2πn:t=0.1arcsin(2519​)+2πn​
0.1t=arcsin(2519​)+2πn
Divide both sides by 0.1
0.1t=arcsin(2519​)+2πn
Divide both sides by 0.10.10.1t​=0.1arcsin(2519​)​+0.12πn​
Simplify
0.10.1t​=0.1arcsin(2519​)​+0.12πn​
Simplify 0.10.1t​:t
0.10.1t​
Cancel the common factor: 0.1=t
Simplify 0.1arcsin(2519​)​+0.12πn​:0.1arcsin(2519​)+2πn​
0.1arcsin(2519​)​+0.12πn​
Apply rule ca​±cb​=ca±b​=0.1arcsin(2519​)+2πn​
t=0.1arcsin(2519​)+2πn​
t=0.1arcsin(2519​)+2πn​
t=0.1arcsin(2519​)+2πn​
0.1t=π−arcsin(2519​)+2πn:t=0.1π−arcsin(2519​)+2πn​
0.1t=π−arcsin(2519​)+2πn
Divide both sides by 0.1
0.1t=π−arcsin(2519​)+2πn
Divide both sides by 0.10.10.1t​=0.1π​−0.1arcsin(2519​)​+0.12πn​
Simplify
0.10.1t​=0.1π​−0.1arcsin(2519​)​+0.12πn​
Simplify 0.10.1t​:t
0.10.1t​
Cancel the common factor: 0.1=t
Simplify 0.1π​−0.1arcsin(2519​)​+0.12πn​:0.1π−arcsin(2519​)+2πn​
0.1π​−0.1arcsin(2519​)​+0.12πn​
Apply rule ca​±cb​=ca±b​=0.1π−arcsin(2519​)+2πn​
t=0.1π−arcsin(2519​)+2πn​
t=0.1π−arcsin(2519​)+2πn​
t=0.1π−arcsin(2519​)+2πn​
t=5π+20πn,t=15π+20πn,t=0.1arcsin(2519​)+2πn​,t=0.1π−arcsin(2519​)+2πn​
Show solutions in decimal formt=5π+20πn,t=15π+20πn,t=0.10.86331…+2πn​,t=0.1π−0.86331…+2πn​

Graph

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

Popular Examples

sin(x/4)=(sqrt(3))/2cos(θ)=(-3+(-6))/((sqrt(9))(\sqrt{18))}sin(6/x)=368cos(2x)+6=cos^2(x)+cos(x)tan(x)= 200/500
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