What is the general solution for tan(4x)*cot(x+60)=1 ?

The general solution for tan(4x)*cot(x+60)=1 is x=(2pin)/3+20,x= pi/3+20+(2pin)/3

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tan(4x)*cot(x+60)=1

Solution

tan (4 x) \cdot cot (x + 60) = 1

Solution

x = \frac{2 πn}{3} + 20, x = \frac{π}{3} + 20 + \frac{2 πn}{3}

Degrees

x = 1145.91559 \dots^{\circ} + 12 0^{\circ} n, x = 1205.91559 \dots^{\circ} + 12 0^{\circ} n

Solution steps

tan (4 x) cot (x + 60) = 1

Subtract

1

from both sides

tan (4 x) cot (x + 60) - 1 = 0

Express with sin, cos

- 1 + cot (60 + x) tan (4 x)

Use the basic trigonometric identity:

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

= - 1 + \frac{cos ( 60 + x )}{sin ( 60 + x )} tan (4 x)

Use the basic trigonometric identity:

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

= - 1 + \frac{cos ( 60 + x )}{sin ( 60 + x )} \cdot \frac{sin ( 4 x )}{cos ( 4 x )}

Simplify

- 1 + \frac{cos ( 60 + x )}{sin ( 60 + x )} \cdot \frac{sin ( 4 x )}{cos ( 4 x )} : \frac{- sin ( 60 + x ) cos ( 4 x ) + cos ( 60 + x ) sin ( 4 x )}{sin ( 60 + x ) cos ( 4 x )}

- 1 + \frac{cos ( 60 + x )}{sin ( 60 + x )} \cdot \frac{sin ( 4 x )}{cos ( 4 x )}

Multiply

\frac{cos ( 60 + x )}{sin ( 60 + x )} \cdot \frac{sin ( 4 x )}{cos ( 4 x )} : \frac{cos ( x + 60 ) sin ( 4 x )}{sin ( x + 60 ) cos ( 4 x )}

\frac{cos ( 60 + x )}{sin ( 60 + x )} \cdot \frac{sin ( 4 x )}{cos ( 4 x )}

Multiply fractions:

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

= \frac{cos ( 60 + x ) sin ( 4 x )}{sin ( 60 + x ) cos ( 4 x )}

= - 1 + \frac{cos ( x + 60 ) sin ( 4 x )}{sin ( x + 60 ) cos ( 4 x )}

Convert element to fraction:

1 = \frac{1 sin ( 60 + x ) cos ( 4 x )}{sin ( 60 + x ) cos ( 4 x )}

= - \frac{1 \cdot sin ( 60 + x ) cos ( 4 x )}{sin ( 60 + x ) cos ( 4 x )} + \frac{cos ( 60 + x ) sin ( 4 x )}{sin ( 60 + x ) cos ( 4 x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot sin ( 60 + x ) cos ( 4 x ) + cos ( 60 + x ) sin ( 4 x )}{sin ( 60 + x ) cos ( 4 x )}

Multiply:

1 \cdot sin (60 + x) = sin (60 + x)

= \frac{- sin ( x + 60 ) cos ( 4 x ) + cos ( x + 60 ) sin ( 4 x )}{sin ( x + 60 ) cos ( 4 x )}

= \frac{- sin ( 60 + x ) cos ( 4 x ) + cos ( 60 + x ) sin ( 4 x )}{sin ( 60 + x ) cos ( 4 x )}

\frac{- cos ( 4 x ) sin ( 60 + x ) + cos ( 60 + x ) sin ( 4 x )}{cos ( 4 x ) sin ( 60 + x )} = 0

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

- cos (4 x) sin (60 + x) + cos (60 + x) sin (4 x) = 0

Rewrite using trig identities

- cos (4 x) sin (60 + x) + cos (60 + x) sin (4 x)

Use the Angle Difference identity:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= sin (4 x - (60 + x))

sin (4 x - (60 + x)) = 0

General solutions for

sin (4 x - (60 + x)) = 0

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}

4 x - (60 + x) = 0 + 2 πn, 4 x - (60 + x) = π + 2 πn

4 x - (60 + x) = 0 + 2 πn, 4 x - (60 + x) = π + 2 πn

Solve

4 x - (60 + x) = 0 + 2 πn : x = \frac{2 πn}{3} + 20

4 x - (60 + x) = 0 + 2 πn

0 + 2 πn = 2 πn

4 x - (60 + x) = 2 πn

Expand

4 x - (60 + x) : 3 x - 60

4 x - (60 + x)

- (60 + x) : - 60 - x

- (60 + x)

Distribute parentheses

= - (60) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 60 - x

= 4 x - 60 - x

Simplify

4 x - 60 - x : 3 x - 60

4 x - 60 - x

Group like terms

= 4 x - x - 60

Add similar elements:

4 x - x = 3 x

= 3 x - 60

= 3 x - 60

3 x - 60 = 2 πn

Move

60

to the right side

3 x - 60 = 2 πn

Add

60

to both sides

3 x - 60 + 60 = 2 πn + 60

Simplify

3 x = 2 πn + 60

3 x = 2 πn + 60

Divide both sides by

3

3 x = 2 πn + 60

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{60}{3}

Simplify

x = \frac{2 πn}{3} + 20

x = \frac{2 πn}{3} + 20

Solve

4 x - (60 + x) = π + 2 πn : x = \frac{π}{3} + 20 + \frac{2 πn}{3}

4 x - (60 + x) = π + 2 πn

Expand

4 x - (60 + x) : 3 x - 60

4 x - (60 + x)

- (60 + x) : - 60 - x

- (60 + x)

Distribute parentheses

= - (60) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 60 - x

= 4 x - 60 - x

Simplify

4 x - 60 - x : 3 x - 60

4 x - 60 - x

Group like terms

= 4 x - x - 60

Add similar elements:

4 x - x = 3 x

= 3 x - 60

= 3 x - 60

3 x - 60 = π + 2 πn

Move

60

to the right side

3 x - 60 = π + 2 πn

Add

60

to both sides

3 x - 60 + 60 = π + 2 πn + 60

Simplify

3 x = π + 2 πn + 60

3 x = π + 2 πn + 60

Divide both sides by

3

3 x = π + 2 πn + 60

Divide both sides by

3

\frac{3 x}{3} = \frac{π}{3} + \frac{2 πn}{3} + \frac{60}{3}

Simplify

\frac{3 x}{3} = \frac{π}{3} + \frac{2 πn}{3} + \frac{60}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{π}{3} + \frac{2 πn}{3} + \frac{60}{3} : \frac{π}{3} + 20 + \frac{2 πn}{3}

\frac{π}{3} + \frac{2 πn}{3} + \frac{60}{3}

Group like terms

= \frac{π}{3} + \frac{60}{3} + \frac{2 πn}{3}

Divide the numbers:

\frac{60}{3} = 20

= \frac{π}{3} + 20 + \frac{2 πn}{3}

x = \frac{π}{3} + 20 + \frac{2 πn}{3}

x = \frac{π}{3} + 20 + \frac{2 πn}{3}

x = \frac{π}{3} + 20 + \frac{2 πn}{3}

x = \frac{2 πn}{3} + 20, x = \frac{π}{3} + 20 + \frac{2 πn}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for tan(4x)*cot(x+60)=1 ?
The general solution for tan(4x)*cot(x+60)=1 is x=(2pin)/3+20,x= pi/3+20+(2pin)/3