What is the general solution for cot(θ/7)=sqrt(12) ?

The general solution for cot(θ/7)=sqrt(12) is θ=7*0.28103…+7pin

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cot(θ/7)=sqrt(12)

cot (\frac{θ}{7}) = 12

θ = 7 \cdot 0.28103 \dots + 7 πn

Radians

θ = 1.96724 \dots + 21.99114 \dots n

Solution steps

cot (\frac{θ}{7}) = 12

Simplify

12 : 23

12

Prime factorization of

12 : 2^{2} \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

Apply radical rule:

= 3 2^{2}

Apply radical rule:

2^{2} = 2

= 23

Apply trig inverse properties

cot (\frac{θ}{7}) = 23

General solutions for

cot (\frac{θ}{7}) = 23

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

\frac{θ}{7} = arccot (23) + πn

\frac{θ}{7} = arccot (23) + πn

Solve

\frac{θ}{7} = arccot (23) + πn : θ = 7 arccot (23) + 7 πn

\frac{θ}{7} = arccot (23) + πn

Multiply both sides by

7

\frac{θ}{7} = arccot (23) + πn

Multiply both sides by

7

\frac{7 θ}{7} = 7 arccot (23) + 7 πn

Simplify

θ = 7 arccot (23) + 7 πn

θ = 7 arccot (23) + 7 πn

θ = 7 arccot (23) + 7 πn

Show solutions in decimal form

θ = 7 \cdot 0.28103 \dots + 7 πn

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

2 sin (θ) + 6 = 5

3 sin (θ) - 3 = sin (θ)

cos (x) = \frac{15}{17}

6 cos (θ) - 3 = 0

What is the general solution for cot(θ/7)=sqrt(12) ?
The general solution for cot(θ/7)=sqrt(12) is θ=7*0.28103…+7pin