What is the general solution for cot^2(θ)-9=0 ?

The general solution for cot^2(θ)-9=0 is θ=0.32175…+pin,θ=2.81984…+pin

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cot^2(θ)-9=0

cot^{2} (θ) - 9 = 0

θ = 0.32175 \dots + πn, θ = 2.81984 \dots + πn

Degrees

θ = 18.43494 \dots^{\circ} + 18 0^{\circ} n, θ = 161.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

cot^{2} (θ) - 9 = 0

Solve by substitution

cot^{2} (θ) - 9 = 0

Let:

cot (θ) = u

u^{2} - 9 = 0

u^{2} - 9 = 0 : u = 3, u = - 3

u^{2} - 9 = 0

Move

9

to the right side

u^{2} - 9 = 0

Add

9

to both sides

u^{2} - 9 + 9 = 0 + 9

Simplify

u^{2} = 9

u^{2} = 9

For

x^{2} = f (a)

the solutions are

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

u = 9, u = - 9

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

- 9 = - 3

- 9

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= - 3

u = 3, u = - 3

Substitute back

u = cot (θ)

cot (θ) = 3, cot (θ) = - 3

cot (θ) = 3, cot (θ) = - 3

cot (θ) = 3 : θ = arccot (3) + πn

cot (θ) = 3

Apply trig inverse properties

cot (θ) = 3

General solutions for

cot (θ) = 3

cot (x) = a \Rightarrow x = arccot (a) + πn

θ = arccot (3) + πn

θ = arccot (3) + πn

cot (θ) = - 3 : θ = arccot (- 3) + πn

cot (θ) = - 3

Apply trig inverse properties

cot (θ) = - 3

General solutions for

cot (θ) = - 3

cot (x) = - a \Rightarrow x = arccot (- a) + πn

θ = arccot (- 3) + πn

θ = arccot (- 3) + πn

Combine all the solutions

θ = arccot (3) + πn, θ = arccot (- 3) + πn

Show solutions in decimal form

θ = 0.32175 \dots + πn, θ = 2.81984 \dots + πn

sec (x) = - 1 - tan (x)

sec (θ) + 3 = 0

2 sin (x) cos (x) + 3 sin (x) = 0

cos (2 x) - cos (x) = 0, 0 \leq x < 2 π

x, 2 cos (2 x) - 1 = 0

What is the general solution for cot^2(θ)-9=0 ?
The general solution for cot^2(θ)-9=0 is θ=0.32175…+pin,θ=2.81984…+pin