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

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

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1/(cos(x))=-2

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

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) = 4

sin (θ) = - 0.9

cos (2 T) = 1, 0 \leq T \leq 2 π

2 - 2 cos^{2} (\frac{x}{2}) = 2 cos^{2} (x)

sec^{2} (θ) - 2 = tan^{2} (θ)

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