What is the general solution for sec(x)= 1/(sin(x)) ?

The general solution for sec(x)= 1/(sin(x)) is x= pi/4+pin

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sec(x)= 1/(sin(x))

sec (x) = \frac{1}{sin ( x )}

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

sec (x) = \frac{1}{sin ( x )}

Subtract

\frac{1}{sin ( x )}

from both sides

sec (x) - \frac{1}{sin ( x )} = 0

Simplify

sec (x) - \frac{1}{sin ( x )} : \frac{sec ( x ) sin ( x ) - 1}{sin ( x )}

sec (x) - \frac{1}{sin ( x )}

Convert element to fraction:

sec (x) = \frac{sec ( x ) sin ( x )}{sin ( x )}

= \frac{sec ( x ) sin ( x )}{sin ( x )} - \frac{1}{sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sec ( x ) sin ( x ) - 1}{sin ( x )}

\frac{sec ( x ) sin ( x ) - 1}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sec (x) sin (x) - 1 = 0

Rewrite using trig identities

sec (x) sin (x) - 1

sec (x) sin (x) = tan (x)

sec (x) sin (x)

Express with sin, cos

sec (x) sin (x)

Use the basic trigonometric identity:

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

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

Simplify

\frac{1}{cos ( x )} sin (x) : \frac{sin ( x )}{cos ( x )}

\frac{1}{cos ( x )} sin (x)

Multiply fractions:

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

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

Multiply:

1 \cdot sin (x) = sin (x)

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

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

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

= tan (x)

= - 1 + tan (x)

- 1 + tan (x) = 0

Move

1

to the right side

- 1 + tan (x) = 0

Add

1

to both sides

- 1 + tan (x) + 1 = 0 + 1

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

sin (x) sec (x) = 2 sin (x)

sin (θ) tan (θ) = sin (θ)

sin (5 x) = \frac{1}{2}

tan (t) = - 3

4 cos (θ) - 3 = 0

What is the general solution for sec(x)= 1/(sin(x)) ?
The general solution for sec(x)= 1/(sin(x)) is x= pi/4+pin