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Popular Trigonometry >

(625^{cos^2(x)})/(25^{cos(x))}=1

  • Pre Algebra
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Solution

25cos(x)625cos2(x)​=1

Solution

x=2π​+2πn,x=23π​+2πn,x=3π​+2πn,x=35π​+2πn
+1
Degrees
x=90∘+360∘n,x=270∘+360∘n,x=60∘+360∘n,x=300∘+360∘n
Solution steps
25cos(x)625cos2(x)​=1
Solve by substitution
25cos(x)625cos2(x)​=1
Let: cos(x)=u25u625u2​=1
25u625u2​=1:u=0,u=21​
25u625u2​=1
Apply exponent rules
25u625u2​=1
Apply exponent rule: ab1​=a−b25u1​=25−u625u2⋅25−u=1
Convert to base 25:252u2⋅25−u=1
Convert 625 to base 25625=252(252)u2⋅25−u=1
Apply exponent rule: (ab)c=abc(252)u2=252u2252u2⋅25−u=1
252u2⋅25−u=1
Apply exponent rule: ab⋅ac=ab+c252u2⋅25−u=252u2−u252u2−u=1
If f(x)=g(x), then ln(f(x))=ln(g(x))ln(252u2−u)=ln(1)
Apply log rule: ln(xa)=a⋅ln(x)ln(252u2−u)=(2u2−u)ln(25)(2u2−u)ln(25)=ln(1)
(2u2−u)ln(25)=ln(1)
Solve (2u2−u)ln(25)=ln(1):u=0,u=21​
(2u2−u)ln(25)=ln(1)
Factor (2u2−u)ln(25):2ln(5)u(2u−1)
(2u2−u)ln(25)
Factor 2u2−u:u(2u−1)
2u2−u
Factor out common term u:u(2u−1)
2u2−u
Apply exponent rule: ab+c=abacu2=uu=2uu−u
Factor out common term u=u(2u−1)
=u(2u−1)
=u(2u−1)ln(25)
Simplify ln(25):2ln(5)
ln(25)
Rewrite 25 in power-base form:25=52=ln(52)
Apply log rule: loga​(xb)=b⋅loga​(x)ln(52)=2ln(5)=2ln(5)
=2ln(5)u(2u−1)
2ln(5)u(2u−1)=ln(1)
Using the Zero Factor Principle: If ab=0then a=0or b=0u=0or2u−1=0
Solve 2u−1=0:u=21​
2u−1=0
Move 1to the right side
2u−1=0
Add 1 to both sides2u−1+1=0+1
Simplify2u=1
2u=1
Divide both sides by 2
2u=1
Divide both sides by 222u​=21​
Simplifyu=21​
u=21​
The solutions to the quadratic equation are:u=0,u=21​
u=0,u=21​
Verify Solutions:u=0True,u=21​True
Check the solutions by plugging them into 25u625u2​=1
Remove the ones that don't agree with the equation.
Plug in u=0:True
25062502​=1
25062502​=1
25062502​
Apply rule 0a=002=0=2506250​
Apply rule a0=1,a=0250=1=16250​
Apply rule a0=1,a=0=11​
Apply rule 1a​=a=1
1=1
True
Plug in u=21​:True
25(21​)625(21​)2​=1
25(21​)625(21​)2​=1
2521​625(21​)2​
2521​=5
2521​
Factor the number: 25=52=(52)21​
Apply exponent rule: (ab)c=abc(52)21​=52⋅21​=5=5
=5625(21​)2​
625(21​)2=625221​
625(21​)2
(21​)2=221​
(21​)2
Apply exponent rule: (ba​)c=bcac​=2212​
Apply rule 1a=112=1=221​
=625221​
=5625221​​
22=4=562541​​
62541​=5
62541​
Factor the number: 625=54=(54)41​
Apply exponent rule: (ab)c=abc(54)41​=54⋅41​=5=5
=55​
Apply rule aa​=1=1
1=1
True
The solutions areu=0,u=21​
Substitute back u=cos(x)cos(x)=0,cos(x)=21​
cos(x)=0,cos(x)=21​
cos(x)=0:x=2π​+2πn,x=23π​+2πn
cos(x)=0
General solutions for cos(x)=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​​​​
x=2π​+2πn,x=23π​+2πn
x=2π​+2πn,x=23π​+2πn
cos(x)=21​:x=3π​+2πn,x=35π​+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=3π​+2πn,x=35π​+2πn
x=3π​+2πn,x=35π​+2πn
Combine all the solutionsx=2π​+2πn,x=23π​+2πn,x=3π​+2πn,x=35π​+2πn

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