What is the general solution for (cot(θ)-1)(2sin(θ)+sqrt(3))=0 ?

The general solution for (cot(θ)-1)(2sin(θ)+sqrt(3))=0 is θ= pi/4+pin,θ=(4pi)/3+2pin,θ=(5pi)/3+2pin

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(cot(θ)-1)(2sin(θ)+sqrt(3))=0

Solution

(cot (θ) - 1) (2 sin (θ) + 3) = 0

Solution

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

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n, θ = 24 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

(cot (θ) - 1) (2 sin (θ) + 3) = 0

Solving each part separately

cot (θ) - 1 = 0 or 2 sin (θ) + 3 = 0

cot (θ) - 1 = 0 : θ = \frac{π}{4} + πn

cot (θ) - 1 = 0

Move

1

to the right side

cot (θ) - 1 = 0

Add

1

to both sides

cot (θ) - 1 + 1 = 0 + 1

Simplify

cot (θ) = 1

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

2 sin (θ) + 3 = 0 : θ = \frac{4 π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

2 sin (θ) + 3 = 0

Move

3

to the right side

2 sin (θ) + 3 = 0

Subtract

3

from both sides

2 sin (θ) + 3 - 3 = 0 - 3

Simplify

2 sin (θ) = - 3

2 sin (θ) = - 3

Divide both sides by

2

2 sin (θ) = - 3

Divide both sides by

2

\frac{2 sin ( θ )}{2} = \frac{- 3}{2}

Simplify

sin (θ) = - \frac{3}{2}

sin (θ) = - \frac{3}{2}

General solutions for

sin (θ) = - \frac{3}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Combine all the solutions

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

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1 = - cot^{2} (x) + 2

60*pi/(180)=2arctan(x/(17))

60 \cdot \frac{π}{180} = 2 arctan (\frac{x}{17})

sin(a)=-1/8 ,(3pi)/2 <= a<= 2pi

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3cos^2(x)-2sqrt(3)cos(x)=sin^2(x)

3 cos^{2} (x) - 23 cos (x) = sin^{2} (x)

Frequently Asked Questions (FAQ)

What is the general solution for (cot(θ)-1)(2sin(θ)+sqrt(3))=0 ?
The general solution for (cot(θ)-1)(2sin(θ)+sqrt(3))=0 is θ= pi/4+pin,θ=(4pi)/3+2pin,θ=(5pi)/3+2pin