What is the general solution for csc^2(θ)-2=0 ?

The general solution for csc^2(θ)-2=0 is θ= pi/4+2pin,θ=(3pi)/4+2pin,θ=(5pi)/4+2pin,θ=(7pi)/4+2pin

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csc^2(θ)-2=0

csc^{2} (θ) - 2 = 0

θ = \frac{π}{4} + 2 πn, θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

Degrees

θ = 4 5^{\circ} + 36 0^{\circ} n, θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 22 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

csc^{2} (θ) - 2 = 0

Solve by substitution

csc^{2} (θ) - 2 = 0

Let:

csc (θ) = u

u^{2} - 2 = 0

u^{2} - 2 = 0 : u = 2, u = - 2

u^{2} - 2 = 0

Move

2

to the right side

u^{2} - 2 = 0

Add

2

to both sides

u^{2} - 2 + 2 = 0 + 2

Simplify

u^{2} = 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 2, u = - 2

Substitute back

u = csc (θ)

csc (θ) = 2, csc (θ) = - 2

csc (θ) = 2, csc (θ) = - 2

csc (θ) = 2 : θ = \frac{π}{4} + 2 πn, θ = \frac{3 π}{4} + 2 πn

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}{4} + 2 πn, θ = \frac{3 π}{4} + 2 πn

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

csc (θ) = - 2 : θ = \frac{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

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{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

θ = \frac{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

Combine all the solutions

θ = \frac{π}{4} + 2 πn, θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

2 sin^{2} (x) + 11 cos (x) = 7

sin (A) = \frac{5}{13}

2 cos (x) = - 2

3 tan (3 x) = 3

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

What is the general solution for csc^2(θ)-2=0 ?
The general solution for csc^2(θ)-2=0 is θ= pi/4+2pin,θ=(3pi)/4+2pin,θ=(5pi)/4+2pin,θ=(7pi)/4+2pin