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Popular Trigonometry >

cos(x)-1=sqrt(3)sin(x)

  • Pre Algebra
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Solution

cos(x)−1=3​sin(x)

Solution

x=2πn,x=34π​+2πn
+1
Degrees
x=0∘+360∘n,x=240∘+360∘n
Solution steps
cos(x)−1=3​sin(x)
Square both sides(cos(x)−1)2=(3​sin(x))2
Subtract (3​sin(x))2 from both sides(cos(x)−1)2−3sin2(x)=0
Rewrite using trig identities
(−1+cos(x))2−3sin2(x)
Use the Pythagorean identity: cos2(x)+sin2(x)=1sin2(x)=1−cos2(x)=(−1+cos(x))2−3(1−cos2(x))
Simplify (−1+cos(x))2−3(1−cos2(x)):4cos2(x)−2cos(x)−2
(−1+cos(x))2−3(1−cos2(x))
(−1+cos(x))2:1−2cos(x)+cos2(x)
Apply Perfect Square Formula: (a+b)2=a2+2ab+b2a=−1,b=cos(x)
=(−1)2+2(−1)cos(x)+cos2(x)
Simplify (−1)2+2(−1)cos(x)+cos2(x):1−2cos(x)+cos2(x)
(−1)2+2(−1)cos(x)+cos2(x)
Remove parentheses: (−a)=−a=(−1)2−2⋅1⋅cos(x)+cos2(x)
(−1)2=1
(−1)2
Apply exponent rule: (−a)n=an,if n is even(−1)2=12=12
Apply rule 1a=1=1
2⋅1⋅cos(x)=2cos(x)
2⋅1⋅cos(x)
Multiply the numbers: 2⋅1=2=2cos(x)
=1−2cos(x)+cos2(x)
=1−2cos(x)+cos2(x)
=1−2cos(x)+cos2(x)−3(1−cos2(x))
Expand −3(1−cos2(x)):−3+3cos2(x)
−3(1−cos2(x))
Apply the distributive law: a(b−c)=ab−aca=−3,b=1,c=cos2(x)=−3⋅1−(−3)cos2(x)
Apply minus-plus rules−(−a)=a=−3⋅1+3cos2(x)
Multiply the numbers: 3⋅1=3=−3+3cos2(x)
=1−2cos(x)+cos2(x)−3+3cos2(x)
Simplify 1−2cos(x)+cos2(x)−3+3cos2(x):4cos2(x)−2cos(x)−2
1−2cos(x)+cos2(x)−3+3cos2(x)
Group like terms=−2cos(x)+cos2(x)+3cos2(x)+1−3
Add similar elements: cos2(x)+3cos2(x)=4cos2(x)=−2cos(x)+4cos2(x)+1−3
Add/Subtract the numbers: 1−3=−2=4cos2(x)−2cos(x)−2
=4cos2(x)−2cos(x)−2
=4cos2(x)−2cos(x)−2
−2−2cos(x)+4cos2(x)=0
Solve by substitution
−2−2cos(x)+4cos2(x)=0
Let: cos(x)=u−2−2u+4u2=0
−2−2u+4u2=0:u=1,u=−21​
−2−2u+4u2=0
Write in the standard form ax2+bx+c=04u2−2u−2=0
Solve with the quadratic formula
4u2−2u−2=0
Quadratic Equation Formula:
For a=4,b=−2,c=−2u1,2​=2⋅4−(−2)±(−2)2−4⋅4(−2)​​
u1,2​=2⋅4−(−2)±(−2)2−4⋅4(−2)​​
(−2)2−4⋅4(−2)​=6
(−2)2−4⋅4(−2)​
Apply rule −(−a)=a=(−2)2+4⋅4⋅2​
Apply exponent rule: (−a)n=an,if n is even(−2)2=22=22+4⋅4⋅2​
Multiply the numbers: 4⋅4⋅2=32=22+32​
22=4=4+32​
Add the numbers: 4+32=36=36​
Factor the number: 36=62=62​
Apply radical rule: 62​=6=6
u1,2​=2⋅4−(−2)±6​
Separate the solutionsu1​=2⋅4−(−2)+6​,u2​=2⋅4−(−2)−6​
u=2⋅4−(−2)+6​:1
2⋅4−(−2)+6​
Apply rule −(−a)=a=2⋅42+6​
Add the numbers: 2+6=8=2⋅48​
Multiply the numbers: 2⋅4=8=88​
Apply rule aa​=1=1
u=2⋅4−(−2)−6​:−21​
2⋅4−(−2)−6​
Apply rule −(−a)=a=2⋅42−6​
Subtract the numbers: 2−6=−4=2⋅4−4​
Multiply the numbers: 2⋅4=8=8−4​
Apply the fraction rule: b−a​=−ba​=−84​
Cancel the common factor: 4=−21​
The solutions to the quadratic equation are:u=1,u=−21​
Substitute back u=cos(x)cos(x)=1,cos(x)=−21​
cos(x)=1,cos(x)=−21​
cos(x)=1:x=2πn
cos(x)=1
General solutions for cos(x)=1
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​​​​
x=0+2πn
x=0+2πn
Solve x=0+2πn:x=2πn
x=0+2πn
0+2πn=2πnx=2πn
x=2πn
cos(x)=−21​:x=32π​+2πn,x=34π​+2πn
cos(x)=−21​
General solutions for cos(x)=−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​​​​
x=32π​+2πn,x=34π​+2πn
x=32π​+2πn,x=34π​+2πn
Combine all the solutionsx=2πn,x=32π​+2πn,x=34π​+2πn
Verify solutions by plugging them into the original equation
Check the solutions by plugging them into cos(x)−1=3​sin(x)
Remove the ones that don't agree with the equation.
Check the solution 2πn:True
2πn
Plug in n=12π1
For cos(x)−1=3​sin(x)plug inx=2π1cos(2π1)−1=3​sin(2π1)
Refine0=0
⇒True
Check the solution 32π​+2πn:False
32π​+2πn
Plug in n=132π​+2π1
For cos(x)−1=3​sin(x)plug inx=32π​+2π1cos(32π​+2π1)−1=3​sin(32π​+2π1)
Refine−1.5=1.5
⇒False
Check the solution 34π​+2πn:True
34π​+2πn
Plug in n=134π​+2π1
For cos(x)−1=3​sin(x)plug inx=34π​+2π1cos(34π​+2π1)−1=3​sin(34π​+2π1)
Refine−1.5=−1.5
⇒True
x=2πn,x=34π​+2πn

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Frequently Asked Questions (FAQ)

  • What is the general solution for cos(x)-1=sqrt(3)sin(x) ?

    The general solution for cos(x)-1=sqrt(3)sin(x) is x=2pin,x=(4pi)/3+2pin
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