What is the general solution for 1/(cos(x))= 5/3 ?

The general solution for 1/(cos(x))= 5/3 is x=0.92729…+2pin,x=2pi-0.92729…+2pin

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1/(cos(x))= 5/3

\frac{1}{cos ( x )} = \frac{5}{3}

x = 0.92729 \dots + 2 πn, x = 2 π - 0.92729 \dots + 2 πn

Degrees

x = 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 306.86989 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{cos ( x )} = \frac{5}{3}

Cross multiply

\frac{1}{cos ( x )} = \frac{5}{3}

Apply fraction cross multiply: if

\frac{a}{b} = \frac{c}{d}

then

a \cdot d = b \cdot c

1 \cdot 3 = cos (x) \cdot 5

Simplify

1 \cdot 3 : 3

1 \cdot 3

Multiply the numbers:

1 \cdot 3 = 3

= 3

3 = cos (x) \cdot 5

3 = cos (x) \cdot 5

Switch sides

cos (x) \cdot 5 = 3

Divide both sides by

5

cos (x) \cdot 5 = 3

Divide both sides by

5

\frac{cos ( x ) \cdot 5}{5} = \frac{3}{5}

Simplify

cos (x) = \frac{3}{5}

cos (x) = \frac{3}{5}

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{3}{5}

Apply trig inverse properties

cos (x) = \frac{3}{5}

General solutions for

cos (x) = \frac{3}{5}

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

x = arccos (\frac{3}{5}) + 2 πn, x = 2 π - arccos (\frac{3}{5}) + 2 πn

x = arccos (\frac{3}{5}) + 2 πn, x = 2 π - arccos (\frac{3}{5}) + 2 πn

Show solutions in decimal form

x = 0.92729 \dots + 2 πn, x = 2 π - 0.92729 \dots + 2 πn

6 cot (θ) + 5 = 0

cos (x) (2 cos (x) - 1) = 0

cot (θ) = \frac{5}{7}

2 cos (x) - 3 sin (x) = 2

- 2 = 2 sin (x)

What is the general solution for 1/(cos(x))= 5/3 ?
The general solution for 1/(cos(x))= 5/3 is x=0.92729…+2pin,x=2pi-0.92729…+2pin