What is the general solution for sec(θ)csc(θ)=2csc(θ) ?

The general solution for sec(θ)csc(θ)=2csc(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sec(θ)csc(θ)=2csc(θ)

sec (θ) csc (θ) = 2 csc (θ)

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

sec (θ) csc (θ) = 2 csc (θ)

Subtract

2 csc (θ)

from both sides

sec (θ) csc (θ) - 2 csc (θ) = 0

Factor

sec (θ) csc (θ) - 2 csc (θ) : csc (θ) (sec (θ) - 2)

sec (θ) csc (θ) - 2 csc (θ)

Factor out common term

csc (θ)

= csc (θ) (sec (θ) - 2)

csc (θ) (sec (θ) - 2) = 0

Solving each part separately

csc (θ) = 0 or sec (θ) - 2 = 0

csc (θ) = 0 :

No Solution

csc (θ) = 0

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

No Solution

sec (θ) - 2 = 0 : θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

sec (θ) - 2 = 0

Move

2

to the right side

sec (θ) - 2 = 0

Add

2

to both sides

sec (θ) - 2 + 2 = 0 + 2

Simplify

sec (θ) = 2

sec (θ) = 2

General solutions for

sec (θ) = 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Combine all the solutions

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

sin (A) = \frac{( 60 ) sin ( 15 5 ^{\circ} )}{97.73}

cos (θ) = 1 - cos (θ)

tan (θ) = - 1.5

tan (θ) = \frac{30}{20}

10 cos (θ) - 53 = 0

What is the general solution for sec(θ)csc(θ)=2csc(θ) ?
The general solution for sec(θ)csc(θ)=2csc(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin