What is the general solution for 1/(cos(x))=3 ?

The general solution for 1/(cos(x))=3 is x=1.23095…+2pin,x=2pi-1.23095…+2pin

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1/(cos(x))=3

\frac{1}{cos ( x )} = 3

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

Degrees

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{cos ( x )} = 3

Multiply both sides by

cos (x)

\frac{1}{cos ( x )} = 3

Multiply both sides by

cos (x)

\frac{1}{cos ( x )} cos (x) = 3 cos (x)

Simplify

1 = 3 cos (x)

1 = 3 cos (x)

Switch sides

3 cos (x) = 1

Divide both sides by

3

3 cos (x) = 1

Divide both sides by

3

\frac{3 cos ( x )}{3} = \frac{1}{3}

Simplify

cos (x) = \frac{1}{3}

cos (x) = \frac{1}{3}

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}{3}

Apply trig inverse properties

cos (x) = \frac{1}{3}

General solutions for

cos (x) = \frac{1}{3}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Show solutions in decimal form

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

2 sin (x) cos (x) = \frac{3}{2}

sin (2 θ) = - \frac{2 6}{7}, π \leq θ \leq \frac{3 π}{2}

x, cos (x) = \frac{3}{2}

cos (θ) = - \frac{3}{2}, 0 \leq θ \leq 2 π

cos (2 t) = \frac{3}{2}

What is the general solution for 1/(cos(x))=3 ?
The general solution for 1/(cos(x))=3 is x=1.23095…+2pin,x=2pi-1.23095…+2pin