What is the general solution for sin(θ)cot(θ)=(sqrt(3))/2 ?

The general solution for sin(θ)cot(θ)=(sqrt(3))/2 is θ= pi/6+2pin,θ=(11pi)/6+2pin

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sin(θ)cot(θ)=(sqrt(3))/2

sin (θ) cot (θ) = \frac{3}{2}

θ = \frac{π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Degrees

θ = 3 0^{\circ} + 36 0^{\circ} n, θ = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (θ) cot (θ) = \frac{3}{2}

Subtract

\frac{3}{2}

from both sides

sin (θ) cot (θ) - \frac{3}{2} = 0

Simplify

sin (θ) cot (θ) - \frac{3}{2} : \frac{2 sin ( θ ) cot ( θ ) - 3}{2}

sin (θ) cot (θ) - \frac{3}{2}

Convert element to fraction:

sin (θ) cot (θ) = \frac{sin ( θ ) cot ( θ ) 2}{2}

= \frac{sin ( θ ) cot ( θ ) \cdot 2}{2} - \frac{3}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( θ ) cot ( θ ) \cdot 2 - 3}{2}

\frac{2 sin ( θ ) cot ( θ ) - 3}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin (θ) cot (θ) - 3 = 0

Rewrite using trig identities

- 3 + 2 cot (θ) sin (θ)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - 3 + 2 \cdot \frac{cos ( θ )}{sin ( θ )} sin (θ)

2 \cdot \frac{cos ( θ )}{sin ( θ )} sin (θ) = 2 cos (θ)

2 \cdot \frac{cos ( θ )}{sin ( θ )} sin (θ)

Multiply fractions:

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

= \frac{cos ( θ ) \cdot 2 sin ( θ )}{sin ( θ )}

Cancel the common factor:

sin (θ)

= cos (θ) \cdot 2

= - 3 + 2 cos (θ)

- 3 + 2 cos (θ) = 0

Move

3

to the right side

- 3 + 2 cos (θ) = 0

Add

3

to both sides

- 3 + 2 cos (θ) + 3 = 0 + 3

Simplify

2 cos (θ) = 3

2 cos (θ) = 3

Divide both sides by

2

2 cos (θ) = 3

Divide both sides by

2

\frac{2 cos ( θ )}{2} = \frac{3}{2}

Simplify

cos (θ) = \frac{3}{2}

cos (θ) = \frac{3}{2}

General solutions for

cos (θ) = \frac{3}{2}

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

θ = \frac{π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

θ = \frac{π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

cos^{2} (θ) + \frac{1 0 ^{2}}{9.81 ( 1 )} cos (θ) - 1 = 0

cos^{4} (x) = \frac{1}{16}

w, P = 4 (\frac{Y}{P sin ( w )})

tan (2 θ) = \frac{500}{350 - 75}

sec (a) = - 2

What is the general solution for sin(θ)cot(θ)=(sqrt(3))/2 ?
The general solution for sin(θ)cot(θ)=(sqrt(3))/2 is θ= pi/6+2pin,θ=(11pi)/6+2pin