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 >

arcsin(x)-arccos(x)=arcsin(1/2)

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

Solution

arcsin(x)−arccos(x)=arcsin(21​)

Solution

x=23​​
Solution steps
arcsin(x)−arccos(x)=arcsin(21​)
a=b⇒sin(a)=sin(b)sin(arcsin(x)−arccos(x))=sin(arcsin(21​))
Use the following identity: sin(s−t)=sin(s)cos(t)−cos(s)sin(t)sin(arcsin(x))cos(arccos(x))−cos(arcsin(x))sin(arccos(x))=sin(arcsin(21​))
Use the following identity: sin(arcsin(x))=x
Use the following identity: cos(arccos(x))=x
Use the following identity: cos(arcsin(x))=1−x2​
Use the following identity: sin(arccos(x))=1−x2​
xx−1−x2​1−x2​=21​
Solve xx−1−x2​1−x2​=21​:x=23​​,x=−23​​
xx−1−x2​1−x2​=21​
Multiply both sides by 2xx⋅2−1−x2​1−x2​⋅2=21​⋅2
Simplify2x2−2(1−x2​)2=1
Expand 2x2−2(1−x2​)2:4x2−2
2x2−2(1−x2​)2
(1−x2​)2=1−x2
(1−x2​)2
Apply radical rule: a​=a21​=((1−x2)21​)2
Apply exponent rule: (ab)c=abc=(1−x2)21​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=1−x2
=2x2−2(1−x2)
Expand 2x2−2(1−x2):4x2−2
2x2−2(1−x2)
Expand −2(1−x2):−2+2x2
−2(1−x2)
Apply the distributive law: a(b−c)=ab−aca=−2,b=1,c=x2=−2⋅1−(−2)x2
Apply minus-plus rules−(−a)=a=−2⋅1+2x2
Multiply the numbers: 2⋅1=2=−2+2x2
=2x2−2+2x2
Simplify 2x2−2+2x2:4x2−2
2x2−2+2x2
Group like terms=2x2+2x2−2
Add similar elements: 2x2+2x2=4x2=4x2−2
=4x2−2
=4x2−2
4x2−2=1
Solve 4x2−2=1:x=23​​,x=−23​​
4x2−2=1
Move 2to the right side
4x2−2=1
Add 2 to both sides4x2−2+2=1+2
Simplify4x2=3
4x2=3
Divide both sides by 4
4x2=3
Divide both sides by 444x2​=43​
Simplifyx2=43​
x2=43​
For x2=f(a) the solutions are x=f(a)​,−f(a)​
x=43​​,x=−43​​
43​​=23​​
43​​
Apply radical rule: ba​​=b​a​​,a≥0,b≥0=4​3​​
4​=2
4​
Factor the number: 4=22=22​
Apply radical rule: a2​=a,a≥022​=2=2
=23​​
−43​​=−23​​
−43​​
Apply radical rule: ba​​=b​a​​,a≥0,b≥0=−4​3​​
4​=2
4​
Factor the number: 4=22=22​
Apply radical rule: a2​=a,a≥022​=2=2
=−23​​
x=23​​,x=−23​​
x=23​​,x=−23​​
Verify Solutions:x=23​​True,x=−23​​True
Check the solutions by plugging them into xx−1−x2​1−x2​=21​
Remove the ones that don't agree with the equation.
Plug in x=23​​:True
(23​​)(23​​)−1−(23​​)2​1−(23​​)2​=21​
(23​​)(23​​)−1−(23​​)2​1−(23​​)2​=21​
(23​​)(23​​)−1−(23​​)2​1−(23​​)2​
Remove parentheses: (a)=a=23​​⋅23​​−1−(23​​)2​1−(23​​)2​
23​​⋅23​​=43​
23​​⋅23​​
Multiply fractions: ba​⋅dc​=b⋅da⋅c​=2⋅23​3​​
3​3​=3
3​3​
Apply radical rule: a​a​=a3​3​=3=3
=2⋅23​
Multiply the numbers: 2⋅2=4=43​
1−(23​​)2​1−(23​​)2​=41​
1−(23​​)2​1−(23​​)2​
Apply radical rule: a​a​=a−(23​​)2+1​−(23​​)2+1​=1−(23​​)2=1−(23​​)2
(23​​)2=43​
(23​​)2
Apply exponent rule: (ba​)c=bcac​=22(3​)2​
(3​)2:3
Apply radical rule: a​=a21​=(321​)2
Apply exponent rule: (ab)c=abc=321​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=3
=223​
22=4=43​
=1−43​
Convert element to fraction: 1=41⋅4​=41⋅4​−43​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=41⋅4−3​
1⋅4−3=1
1⋅4−3
Multiply the numbers: 1⋅4=4=4−3
Subtract the numbers: 4−3=1=1
=41​
=43​−41​
Apply rule ca​±cb​=ca±b​=43−1​
Subtract the numbers: 3−1=2=42​
Cancel the common factor: 2=21​
21​=21​
True
Plug in x=−23​​:True
(−23​​)(−23​​)−1−(−23​​)2​1−(−23​​)2​=21​
(−23​​)(−23​​)−1−(−23​​)2​1−(−23​​)2​=21​
(−23​​)(−23​​)−1−(−23​​)2​1−(−23​​)2​
Remove parentheses: (−a)=−a,−(−a)=a=23​​⋅23​​−1−(−23​​)2​1−(−23​​)2​
23​​⋅23​​=43​
23​​⋅23​​
Multiply fractions: ba​⋅dc​=b⋅da⋅c​=2⋅23​3​​
3​3​=3
3​3​
Apply radical rule: a​a​=a3​3​=3=3
=2⋅23​
Multiply the numbers: 2⋅2=4=43​
1−(−23​​)2​1−(−23​​)2​=41​
1−(−23​​)2​1−(−23​​)2​
Apply radical rule: a​a​=a−(−23​​)2+1​−(−23​​)2+1​=1−(−23​​)2=1−(−23​​)2
(−23​​)2=43​
(−23​​)2
Apply exponent rule: (−a)n=an,if n is even(−23​​)2=(23​​)2=(23​​)2
Apply exponent rule: (ba​)c=bcac​=22(3​)2​
(3​)2:3
Apply radical rule: a​=a21​=(321​)2
Apply exponent rule: (ab)c=abc=321​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=3
=223​
22=4=43​
=1−43​
Convert element to fraction: 1=41⋅4​=41⋅4​−43​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=41⋅4−3​
1⋅4−3=1
1⋅4−3
Multiply the numbers: 1⋅4=4=4−3
Subtract the numbers: 4−3=1=1
=41​
=43​−41​
Apply rule ca​±cb​=ca±b​=43−1​
Subtract the numbers: 3−1=2=42​
Cancel the common factor: 2=21​
21​=21​
True
The solutions arex=23​​,x=−23​​
x=23​​,x=−23​​
Verify solutions by plugging them into the original equation
Check the solutions by plugging them into arcsin(x)−arccos(x)=arcsin(21​)
Remove the ones that don't agree with the equation.
Check the solution 23​​:True
23​​
Plug in n=123​​
For arcsin(x)−arccos(x)=arcsin(21​)plug inx=23​​arcsin(23​​)−arccos(23​​)=arcsin(21​)
Refine0.52359…=0.52359…
⇒True
Check the solution −23​​:False
−23​​
Plug in n=1−23​​
For arcsin(x)−arccos(x)=arcsin(21​)plug inx=−23​​arcsin(−23​​)−arccos(−23​​)=arcsin(21​)
Refine−3.66519…=0.52359…
⇒False
x=23​​

Graph

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

Popular Examples

3sin^2(x)+sin(x)-4=02sin(x)+5cos(x)=43cos(x)=2-sin(x)cos(2x)=2-3sin(x)arcsin(x)+arcsin(1-x)=arccos(x)

Frequently Asked Questions (FAQ)

  • What is the general solution for arcsin(x)-arccos(x)=arcsin(1/2) ?

    The general solution for arcsin(x)-arccos(x)=arcsin(1/2) is x=(sqrt(3))/2
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