What is the general solution for cot(θ/3+(3pi)/4)=sqrt(3) ?

The general solution for cot(θ/3+(3pi)/4)=sqrt(3) is θ=3pin-(7pi)/4

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Popular Trigonometry >

cot(θ/3+(3pi)/4)=sqrt(3)

Solution

cot (\frac{θ}{3} + \frac{3 π}{4}) = 3

Solution

θ = 3 πn - \frac{7 π}{4}

Degrees

θ = - 31 5^{\circ} + 54 0^{\circ} n

Solution steps

cot (\frac{θ}{3} + \frac{3 π}{4}) = 3

General solutions for

cot (\frac{θ}{3} + \frac{3 π}{4}) = 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

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

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

Solve

\frac{θ}{3} + \frac{3 π}{4} = \frac{π}{6} + πn : θ = 3 πn - \frac{7 π}{4}

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

Move

\frac{3 π}{4}

to the right side

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

Subtract

\frac{3 π}{4}

from both sides

\frac{θ}{3} + \frac{3 π}{4} - \frac{3 π}{4} = \frac{π}{6} + πn - \frac{3 π}{4}

Simplify

\frac{θ}{3} + \frac{3 π}{4} - \frac{3 π}{4} = \frac{π}{6} + πn - \frac{3 π}{4}

Simplify

\frac{θ}{3} + \frac{3 π}{4} - \frac{3 π}{4} : \frac{θ}{3}

\frac{θ}{3} + \frac{3 π}{4} - \frac{3 π}{4}

Add similar elements:

\frac{3 π}{4} - \frac{3 π}{4} = 0

= \frac{θ}{3}

Simplify

\frac{π}{6} + πn - \frac{3 π}{4} : πn - \frac{7 π}{12}

\frac{π}{6} + πn - \frac{3 π}{4}

Group like terms

= πn + \frac{π}{6} - \frac{3 π}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{3 π}{4} :

multiply the denominator and numerator by

3

\frac{3 π}{4} = \frac{3 π 3}{4 \cdot 3} = \frac{9 π}{12}

= \frac{π 2}{12} - \frac{9 π}{12}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π 2 - 9 π}{12}

Add similar elements:

2 π - 9 π = - 7 π

= \frac{- 7 π}{12}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= πn - \frac{7 π}{12}

\frac{θ}{3} = πn - \frac{7 π}{12}

\frac{θ}{3} = πn - \frac{7 π}{12}

\frac{θ}{3} = πn - \frac{7 π}{12}

Multiply both sides by

3

\frac{θ}{3} = πn - \frac{7 π}{12}

Multiply both sides by

3

\frac{3 θ}{3} = 3 πn - 3 \cdot \frac{7 π}{12}

Simplify

\frac{3 θ}{3} = 3 πn - 3 \cdot \frac{7 π}{12}

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

3 πn - 3 \cdot \frac{7 π}{12} : 3 πn - \frac{7 π}{4}

3 πn - 3 \cdot \frac{7 π}{12}

3 \cdot \frac{7 π}{12} = \frac{7 π}{4}

3 \cdot \frac{7 π}{12}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{7 π 3}{12}

Multiply the numbers:

7 \cdot 3 = 21

= \frac{21 π}{12}

Cancel the common factor:

3

= \frac{7 π}{4}

= 3 πn - \frac{7 π}{4}

θ = 3 πn - \frac{7 π}{4}

θ = 3 πn - \frac{7 π}{4}

θ = 3 πn - \frac{7 π}{4}

θ = 3 πn - \frac{7 π}{4}

Graph

Popular Examples

cos(x)=(tan(x)+cot(x))/(csc(x))

2cot^2(y)+3csc(y)=0

cot(x)=-0.456

4tan^2(θ)-2tan(θ)+3=-7tan(θ)+2

-2=csc(θ)

Frequently Asked Questions (FAQ)

What is the general solution for cot(θ/3+(3pi)/4)=sqrt(3) ?
The general solution for cot(θ/3+(3pi)/4)=sqrt(3) is θ=3pin-(7pi)/4