What is the general solution for solvefor y,t=2cot(60y+30) ?

The general solution for solvefor y,t=2cot(60y+30) is y=(3arccot(t/2))/(180)+3n-1/2

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solvefor y,t=2cot(60y+30)

Solution

solve for

y, t = 2 cot (6 0^{\circ} y + 3 0^{\circ})

Solution

y = \frac{3 arccot ( \frac{t}{2} )}{18 0 ^{\circ}} + 3 n - \frac{1}{2}

Radians

y = \frac{3 arccot ( \frac{t}{2} )}{π} - \frac{1}{2} + 3 n

Solution steps

t = 2 cot (6 0^{\circ} y + 3 0^{\circ})

Switch sides

2 cot (\frac{π}{3} y + \frac{π}{6}) = t

Divide both sides by

2

2 cot (\frac{π}{3} y + \frac{π}{6}) = t

Divide both sides by

2

\frac{2 cot ( \frac{π}{3} y + \frac{π}{6} )}{2} = \frac{t}{2}

Simplify

cot (\frac{π}{3} y + \frac{π}{6}) = \frac{t}{2}

cot (\frac{π}{3} y + \frac{π}{6}) = \frac{t}{2}

Apply trig inverse properties

cot (\frac{π}{3} y + \frac{π}{6}) = \frac{t}{2}

General solutions for

cot (6 0^{\circ} y + 3 0^{\circ}) = \frac{t}{2}

cot (x) = a \Rightarrow x = arccot (a) + 18 0^{\circ} n

\frac{π}{3} y + \frac{π}{6} = arccot (\frac{t}{2}) + πn

\frac{π}{3} y + \frac{π}{6} = arccot (\frac{t}{2}) + πn

Solve

6 0^{\circ} y + 3 0^{\circ} = arccot (\frac{t}{2}) + 18 0^{\circ} n : y = \frac{3 arccot ( \frac{t}{2} )}{18 0 ^{\circ}} + 3 n - \frac{1}{2}

\frac{π}{3} y + \frac{π}{6} = arccot (\frac{t}{2}) + πn

Move

3 0^{\circ}

to the right side

\frac{π}{3} y + \frac{π}{6} = arccot (\frac{t}{2}) + πn

Subtract

3 0^{\circ}

from both sides

\frac{π}{3} y + \frac{π}{6} - \frac{π}{6} = arccot (\frac{t}{2}) + πn - \frac{π}{6}

Simplify

\frac{π}{3} y = arccot (\frac{t}{2}) + πn - \frac{π}{6}

\frac{π}{3} y = arccot (\frac{t}{2}) + πn - \frac{π}{6}

Multiply both sides by

3

\frac{π}{3} y = arccot (\frac{t}{2}) + πn - \frac{π}{6}

Multiply both sides by

3

3 \cdot \frac{π}{3} y = 3 arccot (\frac{t}{2}) + 3 πn - 3 \cdot \frac{π}{6}

Simplify

3 \cdot \frac{π}{3} y = 3 arccot (\frac{t}{2}) + 3 πn - 3 \cdot \frac{π}{6}

Simplify

3 \cdot 6 0^{\circ} y : 18 0^{\circ} y

3 \cdot 6 0^{\circ} y

Multiply fractions:

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

= 18 0^{\circ} y

Cancel the common factor:

3

= y 18 0^{\circ}

Simplify

3 arccot (\frac{t}{2}) + 54 0^{\circ} n - 3 \cdot 3 0^{\circ} : 3 arccot (\frac{t}{2}) + 54 0^{\circ} n - 9 0^{\circ}

3 arccot (\frac{t}{2}) + 3 πn - 3 \cdot \frac{π}{6}

3 \cdot 3 0^{\circ} = 9 0^{\circ}

3 \cdot 3 0^{\circ}

Multiply fractions:

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

= 9 0^{\circ}

Cancel the common factor:

3

= 9 0^{\circ}

= 3 arccot (\frac{t}{2}) + 3 πn - \frac{π}{2}

π y = 3 arccot (\frac{t}{2}) + 3 πn - \frac{π}{2}

π y = 3 arccot (\frac{t}{2}) + 3 πn - \frac{π}{2}

π y = 3 arccot (\frac{t}{2}) + 3 πn - \frac{π}{2}

Divide both sides by

18 0^{\circ}

π y = 3 arccot (\frac{t}{2}) + 3 πn - \frac{π}{2}

Divide both sides by

18 0^{\circ}

\frac{π y}{π} = \frac{3 arccot ( \frac{t}{2} )}{π} + \frac{3 πn}{π} - \frac{\frac{π}{2}}{π}

Simplify

\frac{π y}{π} = \frac{3 arccot ( \frac{t}{2} )}{π} + \frac{3 πn}{π} - \frac{\frac{π}{2}}{π}

Simplify

\frac{18 0 ^{\circ} y}{18 0 ^{\circ}} : y

\frac{18 0 ^{\circ} y}{18 0 ^{\circ}}

Cancel the common factor:

18 0^{\circ}

= y

Simplify

\frac{3 arccot ( \frac{t}{2} )}{18 0 ^{\circ}} + \frac{54 0 ^{\circ} n}{18 0 ^{\circ}} - \frac{9 0 ^{\circ}}{18 0 ^{\circ}} : \frac{3 arccot ( \frac{t}{2} )}{18 0 ^{\circ}} + 3 n - \frac{1}{2}

\frac{3 arccot ( \frac{t}{2} )}{π} + \frac{3 πn}{π} - \frac{\frac{π}{2}}{π}

\frac{54 0 ^{\circ} n}{18 0 ^{\circ}} = 3 n

\frac{54 0 ^{\circ} n}{18 0 ^{\circ}}

Cancel the common factor:

18 0^{\circ}

= 3 n

\frac{9 0 ^{\circ}}{18 0 ^{\circ}} = \frac{1}{2}

\frac{9 0 ^{\circ}}{18 0 ^{\circ}}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{36 0 ^{\circ}}

Cancel the common factor:

18 0^{\circ}

= \frac{1}{2}

= \frac{3 arccot ( \frac{t}{2} )}{π} + 3 n - \frac{1}{2}

y = \frac{3 arccot ( \frac{t}{2} )}{π} + 3 n - \frac{1}{2}

y = \frac{3 arccot ( \frac{t}{2} )}{π} + 3 n - \frac{1}{2}

y = \frac{3 arccot ( \frac{t}{2} )}{π} + 3 n - \frac{1}{2}

y = \frac{3 arccot ( \frac{t}{2} )}{18 0 ^{\circ}} + 3 n - \frac{1}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor y,t=2cot(60y+30) ?
The general solution for solvefor y,t=2cot(60y+30) is y=(3arccot(t/2))/(180)+3n-1/2