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Popular Trigonometry >

sin(2x)=-cos(3x)+sin(4x)

  • Pre Algebra
  • Algebra
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Solution

sin(2x)=−cos(3x)+sin(4x)

Solution

x=6π​+32πn​,x=2π​+32πn​
+1
Degrees
x=30∘+120∘n,x=90∘+120∘n
Solution steps
sin(2x)=−cos(3x)+sin(4x)
Subtract −cos(3x)+sin(4x) from both sidessin(2x)+cos(3x)−sin(4x)=0
Rewrite using trig identities
cos(3x)+sin(2x)−sin(4x)
Use the Sum to Product identity: sin(s)−sin(t)=2sin(2s−t​)cos(2s+t​)=cos(3x)+2sin(22x−4x​)cos(22x+4x​)
2sin(22x−4x​)cos(22x+4x​)=−2cos(3x)sin(x)
2sin(22x−4x​)cos(22x+4x​)
22x−4x​=−x
22x−4x​
Add similar elements: 2x−4x=−2x=2−2x​
Apply the fraction rule: b−a​=−ba​=−22x​
Divide the numbers: 22​=1=−x
=2sin(−x)cos(22x+4x​)
Use the negative angle identity: sin(−x)=−sin(x)=2cos(22x+4x​)(−sin(x))
Remove parentheses: (−a)=−a=−2cos(22x+4x​)sin(x)
22x+4x​=3x
22x+4x​
Add similar elements: 2x+4x=6x=26x​
Divide the numbers: 26​=3=3x
=−2cos(3x)sin(x)
=cos(3x)−2cos(3x)sin(x)
cos(3x)−2cos(3x)sin(x)=0
Factor cos(3x)−2cos(3x)sin(x):−cos(3x)(2sin(x)−1)
cos(3x)−2cos(3x)sin(x)
Factor out common term −cos(3x)=−cos(3x)(−1+2sin(x))
−cos(3x)(2sin(x)−1)=0
Solving each part separatelycos(3x)=0or2sin(x)−1=0
cos(3x)=0:x=6π​+32πn​,x=2π​+32πn​
cos(3x)=0
General solutions for cos(3x)=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​​​​
3x=2π​+2πn,3x=23π​+2πn
3x=2π​+2πn,3x=23π​+2πn
Solve 3x=2π​+2πn:x=6π​+32πn​
3x=2π​+2πn
Divide both sides by 3
3x=2π​+2πn
Divide both sides by 333x​=32π​​+32πn​
Simplify
33x​=32π​​+32πn​
Simplify 33x​:x
33x​
Divide the numbers: 33​=1=x
Simplify 32π​​+32πn​:6π​+32πn​
32π​​+32πn​
32π​​=6π​
32π​​
Apply the fraction rule: acb​​=c⋅ab​=2⋅3π​
Multiply the numbers: 2⋅3=6=6π​
=6π​+32πn​
x=6π​+32πn​
x=6π​+32πn​
x=6π​+32πn​
Solve 3x=23π​+2πn:x=2π​+32πn​
3x=23π​+2πn
Divide both sides by 3
3x=23π​+2πn
Divide both sides by 333x​=323π​​+32πn​
Simplify
33x​=323π​​+32πn​
Simplify 33x​:x
33x​
Divide the numbers: 33​=1=x
Simplify 323π​​+32πn​:2π​+32πn​
323π​​+32πn​
323π​​=2π​
323π​​
Apply the fraction rule: acb​​=c⋅ab​=2⋅33π​
Multiply the numbers: 2⋅3=6=63π​
Cancel the common factor: 3=2π​
=2π​+32πn​
x=2π​+32πn​
x=2π​+32πn​
x=2π​+32πn​
x=6π​+32πn​,x=2π​+32πn​
2sin(x)−1=0:x=6π​+2πn,x=65π​+2πn
2sin(x)−1=0
Move 1to the right side
2sin(x)−1=0
Add 1 to both sides2sin(x)−1+1=0+1
Simplify2sin(x)=1
2sin(x)=1
Divide both sides by 2
2sin(x)=1
Divide both sides by 222sin(x)​=21​
Simplifysin(x)=21​
sin(x)=21​
General solutions for sin(x)=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​​​
x=6π​+2πn,x=65π​+2πn
x=6π​+2πn,x=65π​+2πn
Combine all the solutionsx=6π​+32πn​,x=2π​+32πn​,x=6π​+2πn,x=65π​+2πn
Merge Overlapping Intervalsx=6π​+32πn​,x=2π​+32πn​

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Popular Examples

sin(x+pi/2)=-cos(x)sin(x)=(3/5)sin(x)+cos(x)=0.2sin(a)=0.3sin(x)= 5/13 ,tan(x), pi/2 <= x<= pi

Frequently Asked Questions (FAQ)

  • What is the general solution for sin(2x)=-cos(3x)+sin(4x) ?

    The general solution for sin(2x)=-cos(3x)+sin(4x) is x= pi/6+(2pin)/3 ,x= pi/2+(2pin)/3
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