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Popular Trigonometry >

cos(4x)=sin(x)

  • Pre Algebra
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Solution

cos(4x)=sin(x)

Solution

x=10π+4πn​,x=−6π+4πn​
+1
Degrees
x=18∘+72∘n,x=−30∘−120∘n
Solution steps
cos(4x)=sin(x)
Rewrite using trig identities
cos(4x)=sin(x)
Use the following identity: cos(x)=sin(2π​−x)cos(4x)=sin(2π​−4x)
cos(4x)=sin(2π​−4x)
Apply trig inverse properties
cos(4x)=sin(2π​−4x)
sin(x)=sin(y)⇒x=y+2πn,x=π−y+2πnx=2π​−4x+2πn,x=π−(2π​−4x)+2πn
x=2π​−4x+2πn,x=π−(2π​−4x)+2πn
x=2π​−4x+2πn:x=10π+4πn​
x=2π​−4x+2πn
Move 4xto the left side
x=2π​−4x+2πn
Add 4x to both sidesx+4x=2π​−4x+2πn+4x
Simplify5x=2π​+2πn
5x=2π​+2πn
Divide both sides by 5
5x=2π​+2πn
Divide both sides by 555x​=52π​​+52πn​
Simplify
55x​=52π​​+52πn​
Simplify 55x​:x
55x​
Divide the numbers: 55​=1=x
Simplify 52π​​+52πn​:10π+4πn​
52π​​+52πn​
Apply rule ca​±cb​=ca±b​=52π​+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​
=52π+4πn​​
Apply the fraction rule: acb​​=c⋅ab​=2⋅5π+4πn​
Multiply the numbers: 2⋅5=10=10π+4πn​
x=10π+4πn​
x=10π+4πn​
x=10π+4πn​
x=π−(2π​−4x)+2πn:x=−6π+4πn​
x=π−(2π​−4x)+2πn
Expand π−(2π​−4x)+2πn:π−2π​+4x+2πn
π−(2π​−4x)+2πn
−(2π​−4x):−2π​+4x
−(2π​−4x)
Distribute parentheses=−(2π​)−(−4x)
Apply minus-plus rules−(−a)=a,−(a)=−a=−2π​+4x
=π−2π​+4x+2πn
x=π−2π​+4x+2πn
Move 4xto the left side
x=π−2π​+4x+2πn
Subtract 4x from both sidesx−4x=π−2π​+4x+2πn−4x
Simplify−3x=π−2π​+2πn
−3x=π−2π​+2πn
Divide both sides by −3
−3x=π−2π​+2πn
Divide both sides by −3−3−3x​=−3π​−−32π​​+−32πn​
Simplify
−3−3x​=−3π​−−32π​​+−32πn​
Simplify −3−3x​:x
−3−3x​
Apply the fraction rule: −b−a​=ba​=33x​
Divide the numbers: 33​=1=x
Simplify −3π​−−32π​​+−32πn​:−6π+4πn​
−3π​−−32π​​+−32πn​
Apply rule ca​±cb​=ca±b​=−3π−2π​+2πn​
Apply the fraction rule: −ba​=−ba​=−3π−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​
=−32π+4πn​​
Simplify 32π+4πn​​:6π+4πn​
32π+4πn​​
Apply the fraction rule: acb​​=c⋅ab​=2⋅3π+4πn​
Multiply the numbers: 2⋅3=6=6π+4πn​
=−6π+4πn​
x=−6π+4πn​
x=−6π+4πn​
x=−6π+4πn​
x=10π+4πn​,x=−6π+4πn​
x=10π+4πn​,x=−6π+4πn​

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Popular Examples

cos(x)=1cos(-1)sin(x)=(sqrt(3))/2sin((2pi)/4)cos^2(30)

Frequently Asked Questions (FAQ)

  • What is the general solution for cos(4x)=sin(x) ?

    The general solution for cos(4x)=sin(x) is x=(pi+4pin}{10},x=-\frac{pi+4pin)/6
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