What is the general solution for (csc(θ)-2)(cot(θ)+1)=0 ?

The general solution for (csc(θ)-2)(cot(θ)+1)=0 is θ= pi/6+2pin,θ=(5pi)/6+2pin,θ=(3pi)/4+pin

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(csc(θ)-2)(cot(θ)+1)=0

(csc (θ) - 2) (cot (θ) + 1) = 0

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = \frac{3 π}{4} + πn

Degrees

θ = 3 0^{\circ} + 36 0^{\circ} n, θ = 15 0^{\circ} + 36 0^{\circ} n, θ = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

(csc (θ) - 2) (cot (θ) + 1) = 0

Solving each part separately

csc (θ) - 2 = 0 or cot (θ) + 1 = 0

csc (θ) - 2 = 0 : θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

csc (θ) - 2 = 0

Move

2

to the right side

csc (θ) - 2 = 0

Add

2

to both sides

csc (θ) - 2 + 2 = 0 + 2

Simplify

csc (θ) = 2

csc (θ) = 2

General solutions for

csc (θ) = 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

cot (θ) + 1 = 0 : θ = \frac{3 π}{4} + πn

cot (θ) + 1 = 0

Move

1

to the right side

cot (θ) + 1 = 0

Subtract

1

from both sides

cot (θ) + 1 - 1 = 0 - 1

Simplify

cot (θ) = - 1

cot (θ) = - 1

General solutions for

cot (θ) = - 1

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{3 π}{4} + πn

θ = \frac{3 π}{4} + πn

Combine all the solutions

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = \frac{3 π}{4} + πn

sin (x) = csc (x)

9 sin (x) tan (x) = 10 tan (x)

3 sin^{2} (θ) + 6 = 2 sin^{2} (θ) + 7

10 sin^{2} (4 x) = 5

cosh (x) = 2

What is the general solution for (csc(θ)-2)(cot(θ)+1)=0 ?
The general solution for (csc(θ)-2)(cot(θ)+1)=0 is θ= pi/6+2pin,θ=(5pi)/6+2pin,θ=(3pi)/4+pin