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Popular Trigonometry >

sec(x)=4cos(x)

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Solution

sec(x)=4cos(x)

Solution

x=3π​+2πn,x=35π​+2πn,x=32π​+2πn,x=34π​+2πn
+1
Degrees
x=60∘+360∘n,x=300∘+360∘n,x=120∘+360∘n,x=240∘+360∘n
Solution steps
sec(x)=4cos(x)
Subtract 4cos(x) from both sidessec(x)−4cos(x)=0
Rewrite using trig identities
sec(x)−4cos(x)
Use the basic trigonometric identity: cos(x)=sec(x)1​=sec(x)−4⋅sec(x)1​
4⋅sec(x)1​=sec(x)4​
4⋅sec(x)1​
Multiply fractions: a⋅cb​=ca⋅b​=sec(x)1⋅4​
Multiply the numbers: 1⋅4=4=sec(x)4​
=sec(x)−sec(x)4​
−sec(x)4​+sec(x)=0
Solve by substitution
−sec(x)4​+sec(x)=0
Let: sec(x)=u−u4​+u=0
−u4​+u=0:u=2,u=−2
−u4​+u=0
Multiply both sides by u
−u4​+u=0
Multiply both sides by u−u4​u+uu=0⋅u
Simplify
−u4​u+uu=0⋅u
Simplify −u4​u:−4
−u4​u
Multiply fractions: a⋅cb​=ca⋅b​=−u4u​
Cancel the common factor: u=−4
Simplify uu:u2
uu
Apply exponent rule: ab⋅ac=ab+cuu=u1+1=u1+1
Add the numbers: 1+1=2=u2
Simplify 0⋅u:0
0⋅u
Apply rule 0⋅a=0=0
−4+u2=0
−4+u2=0
−4+u2=0
Solve −4+u2=0:u=2,u=−2
−4+u2=0
Move 4to the right side
−4+u2=0
Add 4 to both sides−4+u2+4=0+4
Simplifyu2=4
u2=4
For x2=f(a) the solutions are x=f(a)​,−f(a)​
u=4​,u=−4​
4​=2
4​
Factor the number: 4=22=22​
Apply radical rule: 22​=2=2
−4​=−2
−4​
4​=2
4​
Factor the number: 4=22=22​
Apply radical rule: 22​=2=2
=−2
u=2,u=−2
u=2,u=−2
Verify Solutions
Find undefined (singularity) points:u=0
Take the denominator(s) of −u4​+u and compare to zero
u=0
The following points are undefinedu=0
Combine undefined points with solutions:
u=2,u=−2
Substitute back u=sec(x)sec(x)=2,sec(x)=−2
sec(x)=2,sec(x)=−2
sec(x)=2:x=3π​+2πn,x=35π​+2πn
sec(x)=2
General solutions for sec(x)=2
sec(x) periodicity table with 2πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​sec(x)1323​​2​2Undefined−2−2​−323​​​xπ67π​45π​34π​23π​35π​47π​611π​​sec(x)−1−323​​−2​−2Undefined22​323​​​​
x=3π​+2πn,x=35π​+2πn
x=3π​+2πn,x=35π​+2πn
sec(x)=−2:x=32π​+2πn,x=34π​+2πn
sec(x)=−2
General solutions for sec(x)=−2
sec(x) periodicity table with 2πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​sec(x)1323​​2​2Undefined−2−2​−323​​​xπ67π​45π​34π​23π​35π​47π​611π​​sec(x)−1−323​​−2​−2Undefined22​323​​​​
x=32π​+2πn,x=34π​+2πn
x=32π​+2πn,x=34π​+2πn
Combine all the solutionsx=3π​+2πn,x=35π​+2πn,x=32π​+2πn,x=34π​+2πn

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Popular Examples

cos(z)=10sin(4x)=cos(3x+13)sin(x)=0.848tan(x)=0.35-5cos(x)=3sin^2(x)

Frequently Asked Questions (FAQ)

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

    The general solution for sec(x)=4cos(x) is x= pi/3+2pin,x=(5pi)/3+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin
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