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

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

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sec(θ)= 1/(sin(θ))

sec (θ) = \frac{1}{sin ( θ )}

θ = \frac{π}{4} + πn

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

sec (θ) = \frac{1}{sin ( θ )}

Subtract

\frac{1}{sin ( θ )}

from both sides

sec (θ) - \frac{1}{sin ( θ )} = 0

Simplify

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

sec (θ) - \frac{1}{sin ( θ )}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

sec (θ) sin (θ) - 1 = 0

Rewrite using trig identities

sec (θ) sin (θ) - 1

sec (θ) sin (θ) = tan (θ)

sec (θ) sin (θ)

Express with sin, cos

sec (θ) sin (θ)

Use the basic trigonometric identity:

sec (θ) = \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )} sin (θ)

Simplify

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

\frac{1}{cos ( θ )} sin (θ)

Multiply fractions:

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

= \frac{1 sin ( θ )}{cos ( θ )}

Multiply:

1 \cdot sin (θ) = sin (θ)

= \frac{sin ( θ )}{cos ( θ )}

= \frac{sin ( θ )}{cos ( θ )}

= \frac{sin ( θ )}{cos ( θ )}

Use the basic trigonometric identity:

\frac{sin ( θ )}{cos ( θ )} = tan (θ)

= tan (θ)

= - 1 + tan (θ)

- 1 + tan (θ) = 0

Move

1

to the right side

- 1 + tan (θ) = 0

Add

1

to both sides

- 1 + tan (θ) + 1 = 0 + 1

Simplify

tan (θ) = 1

tan (θ) = 1

General solutions for

tan (θ) = 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}

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

cos (x) = 1, 0 \leq x < 2 π

sin (t) = \frac{1}{3}

5 csc (x) - 3 = 2

2 cos^{2} (x) + 2 = 4

sin (z) = 3

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