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Popular Trigonometry >

cos(3x)+sin(x)=0

  • Pre Algebra
  • Algebra
  • Pre Calculus
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Solution

cos(3x)+sin(x)=0

Solution

x=4π+4πn​,x=−8π+4πn​
+1
Degrees
x=45∘+180∘n,x=−22.5∘−90∘n
Solution steps
cos(3x)+sin(x)=0
Subtract sin(x) from both sidescos(3x)=−sin(x)
Rewrite using trig identities
cos(3x)=−sin(x)
Use the following identity: −sin(x)=sin(−x)cos(3x)=sin(−(x))
Use the following identity: cos(x)=sin(2π​−x)sin(2π​−3x)=sin(−(x))
sin(2π​−3x)=sin(−(x))
Apply trig inverse properties
sin(2π​−3x)=sin(−(x))
sin(x)=sin(y)⇒x=y+2πn,x=π−y+2πn−(x)=2π​−3x+2πn,−(x)=π−(2π​−3x)+2πn
−(x)=2π​−3x+2πn,−(x)=π−(2π​−3x)+2πn
−(x)=2π​−3x+2πn:x=4π+4πn​
−(x)=2π​−3x+2πn
Expand −(x):−x
−(x)
Remove parentheses: (a)=a=−x
−x=2π​−3x+2πn
Move 3xto the left side
−x=2π​−3x+2πn
Add 3x to both sides−x+3x=2π​−3x+2πn+3x
Simplify2x=2π​+2πn
2x=2π​+2πn
Divide both sides by 2
2x=2π​+2πn
Divide both sides by 222x​=22π​​+22πn​
Simplify
22x​=22π​​+22πn​
Simplify 22x​:x
22x​
Divide the numbers: 22​=1=x
Simplify 22π​​+22πn​:4π+4πn​
22π​​+22πn​
Apply rule ca​±cb​=ca±b​=22π​+2πn​
Join 2π​+2πn:2π+4πn​
2π​+2πn
Convert element to fraction: 2πn=22πn2​=2π​+22πn⋅2​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=2π+2πn⋅2​
Multiply the numbers: 2⋅2=4=2π+4πn​
=22π+4πn​​
Apply the fraction rule: acb​​=c⋅ab​=2⋅2π+4πn​
Multiply the numbers: 2⋅2=4=4π+4πn​
x=4π+4πn​
x=4π+4πn​
x=4π+4πn​
−(x)=π−(2π​−3x)+2πn:x=−8π+4πn​
−(x)=π−(2π​−3x)+2πn
Expand −(x):−x
−(x)
Remove parentheses: (a)=a=−x
Expand π−(2π​−3x)+2πn:π−2π​+3x+2πn
π−(2π​−3x)+2πn
−(2π​−3x):−2π​+3x
−(2π​−3x)
Distribute parentheses=−(2π​)−(−3x)
Apply minus-plus rules−(−a)=a,−(a)=−a=−2π​+3x
=π−2π​+3x+2πn
−x=π−2π​+3x+2πn
Move 3xto the left side
−x=π−2π​+3x+2πn
Subtract 3x from both sides−x−3x=π−2π​+3x+2πn−3x
Simplify−4x=π−2π​+2πn
−4x=π−2π​+2πn
Divide both sides by −4
−4x=π−2π​+2πn
Divide both sides by −4−4−4x​=−4π​−−42π​​+−42πn​
Simplify
−4−4x​=−4π​−−42π​​+−42πn​
Simplify −4−4x​:x
−4−4x​
Apply the fraction rule: −b−a​=ba​=44x​
Divide the numbers: 44​=1=x
Simplify −4π​−−42π​​+−42πn​:−8π+4πn​
−4π​−−42π​​+−42πn​
Apply rule ca​±cb​=ca±b​=−4π−2π​+2πn​
Apply the fraction rule: −ba​=−ba​=−4π−2π​+2πn​
Join π−2π​+2πn:2π+4πn​
π−2π​+2πn
Convert element to fraction: π=2π2​,2πn=22πn2​=2π2​−2π​+22πn⋅2​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=2π2−π+2πn⋅2​
π2−π+2πn⋅2=π+4πn
π2−π+2πn⋅2
Add similar elements: 2π−π=π=π+2⋅2πn
Multiply the numbers: 2⋅2=4=π+4πn
=2π+4πn​
=−42π+4πn​​
Simplify 42π+4πn​​:8π+4πn​
42π+4πn​​
Apply the fraction rule: acb​​=c⋅ab​=2⋅4π+4πn​
Multiply the numbers: 2⋅4=8=8π+4πn​
=−8π+4πn​
x=−8π+4πn​
x=−8π+4πn​
x=−8π+4πn​
x=4π+4πn​,x=−8π+4πn​
x=4π+4πn​,x=−8π+4πn​

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Popular Examples

sin(x)cos(x)cos(2x)= 1/8sec(x)=4cos(x)cos(z)=10sin(4x)=cos(3x+13)sin(x)=0.848

Frequently Asked Questions (FAQ)

  • What is the general solution for cos(3x)+sin(x)=0 ?

    The general solution for cos(3x)+sin(x)=0 is x=(pi+4pin)/4 ,x=-(pi+4pin)/8
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