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Popular Trigonometry >

3+cos(4θ)=2-cos(4θ)

  • Pre Algebra
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Solution

3+cos(4θ)=2−cos(4θ)

Solution

θ=6π​+2πn​,θ=3π​+2πn​
+1
Degrees
θ=30∘+90∘n,θ=60∘+90∘n
Solution steps
3+cos(4θ)=2−cos(4θ)
Solve by substitution
3+cos(4θ)=2−cos(4θ)
Let: cos(4θ)=u3+u=2−u
3+u=2−u:u=−21​
3+u=2−u
Move 3to the right side
3+u=2−u
Subtract 3 from both sides3+u−3=2−u−3
Simplifyu=−u−1
u=−u−1
Move uto the left side
u=−u−1
Add u to both sidesu+u=−u−1+u
Simplify2u=−1
2u=−1
Divide both sides by 2
2u=−1
Divide both sides by 222u​=2−1​
Simplifyu=−21​
u=−21​
Substitute back u=cos(4θ)cos(4θ)=−21​
cos(4θ)=−21​
cos(4θ)=−21​:θ=6π​+2πn​,θ=3π​+2πn​
cos(4θ)=−21​
General solutions for cos(4θ)=−21​
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​​​​
4θ=32π​+2πn,4θ=34π​+2πn
4θ=32π​+2πn,4θ=34π​+2πn
Solve 4θ=32π​+2πn:θ=6π​+2πn​
4θ=32π​+2πn
Divide both sides by 4
4θ=32π​+2πn
Divide both sides by 444θ​=432π​​+42πn​
Simplify
44θ​=432π​​+42πn​
Simplify 44θ​:θ
44θ​
Divide the numbers: 44​=1=θ
Simplify 432π​​+42πn​:6π​+2πn​
432π​​+42πn​
432π​​=6π​
432π​​
Apply the fraction rule: acb​​=c⋅ab​=3⋅42π​
Multiply the numbers: 3⋅4=12=122π​
Cancel the common factor: 2=6π​
42πn​=2πn​
42πn​
Cancel the common factor: 2=2πn​
=6π​+2πn​
θ=6π​+2πn​
θ=6π​+2πn​
θ=6π​+2πn​
Solve 4θ=34π​+2πn:θ=3π​+2πn​
4θ=34π​+2πn
Divide both sides by 4
4θ=34π​+2πn
Divide both sides by 444θ​=434π​​+42πn​
Simplify
44θ​=434π​​+42πn​
Simplify 44θ​:θ
44θ​
Divide the numbers: 44​=1=θ
Simplify 434π​​+42πn​:3π​+2πn​
434π​​+42πn​
434π​​=3π​
434π​​
Apply the fraction rule: acb​​=c⋅ab​=3⋅44π​
Multiply the numbers: 3⋅4=12=124π​
Cancel the common factor: 4=3π​
42πn​=2πn​
42πn​
Cancel the common factor: 2=2πn​
=3π​+2πn​
θ=3π​+2πn​
θ=3π​+2πn​
θ=3π​+2πn​
θ=6π​+2πn​,θ=3π​+2πn​
Combine all the solutionsθ=6π​+2πn​,θ=3π​+2πn​

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Popular Examples

5sec(θ)=4sec(θ)+sqrt(2)4sec(pix)+6=-2solvefor x,y-1=-4+3sin(4x-pi/4)64=81+25-90(cos(C))sin^2(a-pi/8)= 2/3

Frequently Asked Questions (FAQ)

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

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