What is the general solution for cot(θ)= 1/(cot(θ)) ?

The general solution for cot(θ)= 1/(cot(θ)) is θ= pi/4+pin,θ=(3pi)/4+pin

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cot(θ)= 1/(cot(θ))

Solution

cot (θ) = \frac{1}{cot ( θ )}

Solution

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

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n, θ = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

cot (θ) = \frac{1}{cot ( θ )}

Solve by substitution

cot (θ) = \frac{1}{cot ( θ )}

Let:

cot (θ) = u

u = \frac{1}{u}

u = \frac{1}{u} : u = 1, u = - 1

u = \frac{1}{u}

Multiply both sides by

u

u = \frac{1}{u}

Multiply both sides by

u

uu = \frac{1}{u} u

Simplify

uu : u^{2}

uu = \frac{1}{u} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

u^{2} = 1

u^{2} = 1

Solve

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

u = 1, u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{1}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - 1

Substitute back

u = cot (θ)

cot (θ) = 1, cot (θ) = - 1

cot (θ) = 1, cot (θ) = - 1

cot (θ) = 1 : θ = \frac{π}{4} + πn

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{π}{4} + πn

θ = \frac{π}{4} + πn

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

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}{4} + πn, θ = \frac{3 π}{4} + πn

Popular Examples

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2 cot^{2} (x) + 2 csc^{2} (x) = 1 + 4 csc (x)

sin(2θ)=cos(θ),0<= θ<2pi

sin (2 θ) = cos (θ), 0 \leq θ < 2 π

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\frac{3}{2} sec (θ) = 3

4csc(θ)-3=0

4 csc (θ) - 3 = 0

Frequently Asked Questions (FAQ)

What is the general solution for cot(θ)= 1/(cot(θ)) ?
The general solution for cot(θ)= 1/(cot(θ)) is θ= pi/4+pin,θ=(3pi)/4+pin