What is the general solution for 1/(cos(x))=2 ?

The general solution for 1/(cos(x))=2 is x= pi/3+2pin,x=(5pi)/3+2pin

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1/(cos(x))=2

\frac{1}{cos ( x )} = 2

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{cos ( x )} = 2

Multiply both sides by

cos (x)

\frac{1}{cos ( x )} = 2

Multiply both sides by

cos (x)

\frac{1}{cos ( x )} cos (x) = 2 cos (x)

Simplify

1 = 2 cos (x)

1 = 2 cos (x)

Switch sides

2 cos (x) = 1

Divide both sides by

2

2 cos (x) = 1

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{1}{2}

Simplify

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

cos (x) = 0

Take the denominator(s) of

\frac{1}{cos ( x )}

and compare to zero

cos (x) = 0

The following points are undefined

cos (x) = 0

Combine undefined points with solutions:

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 tan^{2} (x) = - 3 sec (x)

tan (θ) + 1 = 2

3 sec (θ) + 1 = 7

sin^{2} (x) = cos (x) + 1

tan (x) cos^{2} (x) - tan (x) = 0

What is the general solution for 1/(cos(x))=2 ?
The general solution for 1/(cos(x))=2 is x= pi/3+2pin,x=(5pi)/3+2pin