What is the general solution for cot(θ)+4csc(θ)=6 ?

The general solution for cot(θ)+4csc(θ)=6 is θ=0.88277…+2pin,θ=2.58911…+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cot(θ)+4csc(θ)=6

cot (θ) + 4 csc (θ) = 6

θ = 0.88277 \dots + 2 πn, θ = 2.58911 \dots + 2 πn

Solution steps

cot (θ) + 4 csc (θ) = 6

Subtract

6

from both sides

cot (θ) + 4 csc (θ) - 6 = 0

Express with sin, cos

\frac{cos ( θ )}{sin ( θ )} + 4 \cdot \frac{1}{sin ( θ )} - 6 = 0

Simplify

\frac{cos ( θ )}{sin ( θ )} + 4 \cdot \frac{1}{sin ( θ )} - 6 : \frac{cos ( θ ) + 4 - 6 sin ( θ )}{sin ( θ )}

\frac{cos ( θ ) + 4 - 6 sin ( θ )}{sin ( θ )} = 0

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

cos (θ) + 4 - 6 sin (θ) = 0

Add

6 sin (θ)

to both sides

cos (θ) + 4 = 6 sin (θ)

Square both sides

(cos (θ) + 4)^{2} = (6 sin (θ))^{2}

Subtract

(6 sin (θ))^{2}

from both sides

(cos (θ) + 4)^{2} - 36 sin^{2} (θ) = 0

Rewrite using trig identities

- 20 + 37 cos^{2} (θ) + 8 cos (θ) = 0

Solve by substitution

cos (θ) = \frac{2 ( 3 21 - 2 )}{37}, cos (θ) = - \frac{2 ( 2 + 3 21 )}{37}

cos (θ) = \frac{2 ( 3 21 - 2 )}{37} : θ = arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn

cos (θ) = - \frac{2 ( 2 + 3 21 )}{37} : θ = arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

Combine all the solutions

θ = arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

Verify solutions by plugging them into the original equation

θ = arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

Show solutions in decimal form

θ = 0.88277 \dots + 2 πn, θ = 2.58911 \dots + 2 πn

sin (x) 2 = - sin (x)

tan^{3} (x) = 1

sin^{2} (x) - 9 cos (x) - 9 = 0

sin (a) = \frac{8}{17}

cot (t) = 1

What is the general solution for cot(θ)+4csc(θ)=6 ?
The general solution for cot(θ)+4csc(θ)=6 is θ=0.88277…+2pin,θ=2.58911…+2pin