What is the general solution for sqrt(3)sec(θ)+sec(θ)cot(θ)=0 ?

The general solution for sqrt(3)sec(θ)+sec(θ)cot(θ)=0 is θ=(5pi)/6+pin

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sqrt(3)sec(θ)+sec(θ)cot(θ)=0

3 sec (θ) + sec (θ) cot (θ) = 0

θ = \frac{5 π}{6} + πn

Degrees

θ = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

3 sec (θ) + sec (θ) cot (θ) = 0

Factor

3 sec (θ) + sec (θ) cot (θ) : sec (θ) (3 + cot (θ))

3 sec (θ) + sec (θ) cot (θ)

Factor out common term

sec (θ)

= sec (θ) (3 + cot (θ))

sec (θ) (3 + cot (θ)) = 0

Solving each part separately

sec (θ) = 0 or 3 + cot (θ) = 0

sec (θ) = 0 :

No Solution

sec (θ) = 0

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

No Solution

3 + cot (θ) = 0 : θ = \frac{5 π}{6} + πn

3 + cot (θ) = 0

Move

3

to the right side

3 + cot (θ) = 0

Subtract

3

from both sides

3 + cot (θ) - 3 = 0 - 3

Simplify

cot (θ) = - 3

cot (θ) = - 3

General solutions for

cot (θ) = - 3

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

θ = \frac{5 π}{6} + πn

θ = \frac{5 π}{6} + πn

Combine all the solutions

θ = \frac{5 π}{6} + πn

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

cos (2 x) = 0.5

cos^{2} (x) = - 1

cos (2 x) = 1 - sin (x)

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

What is the general solution for sqrt(3)sec(θ)+sec(θ)cot(θ)=0 ?
The general solution for sqrt(3)sec(θ)+sec(θ)cot(θ)=0 is θ=(5pi)/6+pin