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Popular Trigonometry >

sin(x)-sqrt(3-3sin^2(x))=0

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Solution

sin(x)−3−3sin2(x)​=0

Solution

x=3π​+2πn,x=32π​+2πn
+1
Degrees
x=60∘+360∘n,x=120∘+360∘n
Solution steps
sin(x)−3−3sin2(x)​=0
Solve by substitution
sin(x)−3−3sin2(x)​=0
Let: sin(x)=uu−3−3u2​=0
u−3−3u2​=0:u=23​​
u−3−3u2​=0
Remove square roots
u−3−3u2​=0
Subtract u from both sidesu−3−3u2​−u=0−u
Simplify−3−3u2​=−u
Square both sides:3−3u2=u2
u−3−3u2​=0
(−3−3u2​)2=(−u)2
Expand (−3−3u2​)2:3−3u2
(−3−3u2​)2
Apply exponent rule: (−a)n=an,if n is even(−3−3u2​)2=(3−3u2​)2=(3−3u2​)2
Apply radical rule: a​=a21​=((3−3u2)21​)2
Apply exponent rule: (ab)c=abc=(3−3u2)21​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=3−3u2
Expand (−u)2:u2
(−u)2
Apply exponent rule: (−a)n=an,if n is even(−u)2=u2=u2
3−3u2=u2
3−3u2=u2
3−3u2=u2
Solve 3−3u2=u2:u=23​​,u=−23​​
3−3u2=u2
Move 3to the right side
3−3u2=u2
Subtract 3 from both sides3−3u2−3=u2−3
Simplify−3u2=u2−3
−3u2=u2−3
Move u2to the left side
−3u2=u2−3
Subtract u2 from both sides−3u2−u2=u2−3−u2
Simplify−4u2=−3
−4u2=−3
Divide both sides by −4
−4u2=−3
Divide both sides by −4−4−4u2​=−4−3​
Simplifyu2=43​
u2=43​
For x2=f(a) the solutions are x=f(a)​,−f(a)​
u=43​​,u=−43​​
43​​=23​​
43​​
Apply radical rule: assuming a≥0,b≥0=4​3​​
4​=2
4​
Factor the number: 4=22=22​
Apply radical rule: 22​=2=2
=23​​
−43​​=−23​​
−43​​
Simplify 43​​:23​​
43​​
Apply radical rule: assuming a≥0,b≥0=4​3​​
4​=2
4​
Factor the number: 4=22=22​
Apply radical rule: 22​=2=2
=23​​
=−23​​
u=23​​,u=−23​​
u=23​​,u=−23​​
Verify Solutions:u=23​​True,u=−23​​False
Check the solutions by plugging them into u−3−3u2​=0
Remove the ones that don't agree with the equation.
Plug in u=23​​:True
(23​​)−3−3(23​​)2​=0
(23​​)−3−3(23​​)2​=0
(23​​)−3−3(23​​)2​
Remove parentheses: (a)=a=23​​−3−3(23​​)2​
3−3(23​​)2​=23​​
3−3(23​​)2​
3(23​​)2=49​
3(23​​)2
(23​​)2=223​
(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​
=3⋅223​
Multiply fractions: a⋅cb​=ca⋅b​=223⋅3​
Multiply the numbers: 3⋅3=9=229​
22=4=49​
=3−49​​
Join 3−49​:43​
3−49​
Convert element to fraction: 3=43⋅4​=43⋅4​−49​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=43⋅4−9​
3⋅4−9=3
3⋅4−9
Multiply the numbers: 3⋅4=12=12−9
Subtract the numbers: 12−9=3=3
=43​
=43​​
Apply radical rule: assuming a≥0,b≥0=4​3​​
4​=2
4​
Factor the number: 4=22=22​
Apply radical rule: 22​=2=2
=23​​
=23​​−23​​
Add similar elements: 23​​−23​​=0=0
0=0
True
Plug in u=−23​​:False
(−23​​)−3−3(−23​​)2​=0
(−23​​)−3−3(−23​​)2​=−3​
(−23​​)−3−3(−23​​)2​
Remove parentheses: (−a)=−a=−23​​−3−3(−23​​)2​
3−3(−23​​)2​=23​​
3−3(−23​​)2​
3(−23​​)2=49​
3(−23​​)2
(−23​​)2=223​
(−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​
=3⋅223​
Multiply fractions: a⋅cb​=ca⋅b​=223⋅3​
Multiply the numbers: 3⋅3=9=229​
22=4=49​
=3−49​​
Join 3−49​:43​
3−49​
Convert element to fraction: 3=43⋅4​=43⋅4​−49​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=43⋅4−9​
3⋅4−9=3
3⋅4−9
Multiply the numbers: 3⋅4=12=12−9
Subtract the numbers: 12−9=3=3
=43​
=43​​
Apply radical rule: assuming a≥0,b≥0=4​3​​
4​=2
4​
Factor the number: 4=22=22​
Apply radical rule: 22​=2=2
=23​​
=−23​​−23​​
Apply rule ca​±cb​=ca±b​=2−3​−3​​
Add similar elements: −3​−3​=−23​=2−23​​
Apply the fraction rule: b−a​=−ba​=−223​​
Divide the numbers: 22​=1=−3​
−3​=0
False
The solution isu=23​​
Substitute back u=sin(x)sin(x)=23​​
sin(x)=23​​
sin(x)=23​​:x=3π​+2πn,x=32π​+2πn
sin(x)=23​​
General solutions for sin(x)=23​​
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​​​
x=3π​+2πn,x=32π​+2πn
x=3π​+2πn,x=32π​+2πn
Combine all the solutionsx=3π​+2πn,x=32π​+2πn

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Popular Examples

4tan^2(x)+21tan(x)-49=02sin(2x+15)=1tan(x)+sqrt(3)=sec(x)16sec^2(θ)-1=0cos(4y)=2cos(2y)-1

Frequently Asked Questions (FAQ)

  • What is the general solution for sin(x)-sqrt(3-3sin^2(x))=0 ?

    The general solution for sin(x)-sqrt(3-3sin^2(x))=0 is x= pi/3+2pin,x=(2pi)/3+2pin
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