What is the general solution for sec(θ)=tan(θ) ?

The general solution for sec(θ)=tan(θ) is No Solution for θ\in\mathbb{R}

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sec(θ)=tan(θ)

sec (θ) = tan (θ)

No Solution for θ \in R

Solution steps

sec (θ) = tan (θ)

Subtract

tan (θ)

from both sides

sec (θ) - tan (θ) = 0

Rewrite using trig identities

sec (θ) - tan (θ)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

= - \frac{sin ( θ )}{\frac{1}{s e c ( θ )}} + sec (θ)

\frac{sin ( θ )}{\frac{1}{s e c ( θ )}} = sin (θ) sec (θ)

\frac{sin ( θ )}{\frac{1}{s e c ( θ )}}

Apply the fraction rule:

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

= \frac{sin ( θ ) sec ( θ )}{1}

Apply rule

\frac{a}{1} = a

= sin (θ) sec (θ)

= - sin (θ) sec (θ) + sec (θ)

sec (θ) - sec (θ) sin (θ) = 0

Factor

sec (θ) - sec (θ) sin (θ) : - sec (θ) (sin (θ) - 1)

sec (θ) - sec (θ) sin (θ)

Factor out common term

- sec (θ)

= - sec (θ) (- 1 + sin (θ))

- sec (θ) (sin (θ) - 1) = 0

Solving each part separately

sec (θ) = 0 or sin (θ) - 1 = 0

sec (θ) = 0 :

No Solution

sec (θ) = 0

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

sin (θ) - 1 = 0 : θ = \frac{π}{2} + 2 πn

sin (θ) - 1 = 0

Move

1

to the right side

sin (θ) - 1 = 0

Add

1

to both sides

sin (θ) - 1 + 1 = 0 + 1

Simplify

sin (θ) = 1

sin (θ) = 1

General solutions for

sin (θ) = 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn

No Solution for θ \in R

- 3 csc (θ) = 2

sin (x) - 2 cos^{2} (x) = - 1

- 2 + 2 cos (x) + 4 cos^{2} (x) = 0

sin (A) = \frac{6.347}{12.7}

sec (x) = 3 cos (x) - tan (x)

What is the general solution for sec(θ)=tan(θ) ?
The general solution for sec(θ)=tan(θ) is No Solution for θ\in\mathbb{R}