What is the general solution for tan(2x+1)=-cot(x+3) ?

The general solution for tan(2x+1)=-cot(x+3) is x=2pin+2+pi/2 ,x=2pin+2+(3pi)/2

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tan(2x+1)=-cot(x+3)

Solution

tan (2 x + 1) = - cot (x + 3)

Solution

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

Degrees

x = 204.59155 \dots^{\circ} + 36 0^{\circ} n, x = 384.59155 \dots^{\circ} + 36 0^{\circ} n

Solution steps

tan (2 x + 1) = - cot (x + 3)

Subtract

- cot (x + 3)

from both sides

tan (2 x + 1) + cot (x + 3) = 0

Express with sin, cos

cot (3 + x) + tan (1 + 2 x)

Use the basic trigonometric identity:

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

= \frac{cos ( 3 + x )}{sin ( 3 + x )} + tan (1 + 2 x)

Use the basic trigonometric identity:

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

= \frac{cos ( 3 + x )}{sin ( 3 + x )} + \frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )}

Simplify

\frac{cos ( 3 + x )}{sin ( 3 + x )} + \frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )} : \frac{cos ( 3 + x ) cos ( 2 x + 1 ) + sin ( 1 + 2 x ) sin ( x + 3 )}{sin ( x + 3 ) cos ( 2 x + 1 )}

\frac{cos ( 3 + x )}{sin ( 3 + x )} + \frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )}

Least Common Multiplier of

sin (3 + x), cos (1 + 2 x) : sin (x + 3) cos (2 x + 1)

sin (3 + x), cos (1 + 2 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (3 + x)

cos (1 + 2 x)

= sin (x + 3) cos (2 x + 1)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin (x + 3) cos (2 x + 1)

For

\frac{cos ( 3 + x )}{sin ( 3 + x )} :

multiply the denominator and numerator by

cos (2 x + 1)

\frac{cos ( 3 + x )}{sin ( 3 + x )} = \frac{cos ( 3 + x ) cos ( 2 x + 1 )}{sin ( 3 + x ) cos ( 2 x + 1 )}

For

\frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )} :

multiply the denominator and numerator by

sin (x + 3)

\frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )} = \frac{sin ( 1 + 2 x ) sin ( x + 3 )}{cos ( 1 + 2 x ) sin ( x + 3 )}

= \frac{cos ( 3 + x ) cos ( 2 x + 1 )}{sin ( 3 + x ) cos ( 2 x + 1 )} + \frac{sin ( 1 + 2 x ) sin ( x + 3 )}{cos ( 1 + 2 x ) sin ( x + 3 )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( 3 + x ) cos ( 2 x + 1 ) + sin ( 1 + 2 x ) sin ( x + 3 )}{sin ( x + 3 ) cos ( 2 x + 1 )}

= \frac{cos ( 3 + x ) cos ( 2 x + 1 ) + sin ( 1 + 2 x ) sin ( x + 3 )}{sin ( x + 3 ) cos ( 2 x + 1 )}

\frac{cos ( 1 + 2 x ) cos ( 3 + x ) + sin ( 1 + 2 x ) sin ( 3 + x )}{cos ( 1 + 2 x ) sin ( 3 + x )} = 0

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

cos (1 + 2 x) cos (3 + x) + sin (1 + 2 x) sin (3 + x) = 0

Rewrite using trig identities

cos (1 + 2 x) cos (3 + x) + sin (1 + 2 x) sin (3 + x)

Use the Angle Difference identity:

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

= cos (1 + 2 x - (3 + x))

cos (1 + 2 x - (3 + x)) = 0

General solutions for

cos (1 + 2 x - (3 + x)) = 0

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}

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn, 1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn, 1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn

Solve

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn : x = 2 πn + 2 + \frac{π}{2}

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn

Move

1

to the right side

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn

Subtract

1

from both sides

1 + 2 x - (3 + x) - 1 = \frac{π}{2} + 2 πn - 1

Simplify

2 x - (3 + x) = \frac{π}{2} + 2 πn - 1

2 x - (3 + x) = \frac{π}{2} + 2 πn - 1

Expand

2 x - (3 + x) : x - 3

2 x - (3 + x)

- (3 + x) : - 3 - x

- (3 + x)

Distribute parentheses

= - (3) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 3 - x

= 2 x - 3 - x

Simplify

2 x - 3 - x : x - 3

2 x - 3 - x

Group like terms

= 2 x - x - 3

Add similar elements:

2 x - x = x

= x - 3

= x - 3

x - 3 = \frac{π}{2} + 2 πn - 1

Move

3

to the right side

x - 3 = \frac{π}{2} + 2 πn - 1

Add

3

to both sides

x - 3 + 3 = \frac{π}{2} + 2 πn - 1 + 3

Simplify

x - 3 + 3 = \frac{π}{2} + 2 πn - 1 + 3

Simplify

x - 3 + 3 : x

x - 3 + 3

Add similar elements:

- 3 + 3 = 0

= x

Simplify

\frac{π}{2} + 2 πn - 1 + 3 : 2 πn + 2 + \frac{π}{2}

\frac{π}{2} + 2 πn - 1 + 3

Add/Subtract the numbers:

- 1 + 3 = 2

= 2 πn + 2 + \frac{π}{2}

x = 2 πn + 2 + \frac{π}{2}

x = 2 πn + 2 + \frac{π}{2}

x = 2 πn + 2 + \frac{π}{2}

Solve

1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn : x = 2 πn + 2 + \frac{3 π}{2}

1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn

Move

1

to the right side

1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn

Subtract

1

from both sides

1 + 2 x - (3 + x) - 1 = \frac{3 π}{2} + 2 πn - 1

Simplify

2 x - (3 + x) = \frac{3 π}{2} + 2 πn - 1

2 x - (3 + x) = \frac{3 π}{2} + 2 πn - 1

Expand

2 x - (3 + x) : x - 3

2 x - (3 + x)

- (3 + x) : - 3 - x

- (3 + x)

Distribute parentheses

= - (3) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 3 - x

= 2 x - 3 - x

Simplify

2 x - 3 - x : x - 3

2 x - 3 - x

Group like terms

= 2 x - x - 3

Add similar elements:

2 x - x = x

= x - 3

= x - 3

x - 3 = \frac{3 π}{2} + 2 πn - 1

Move

3

to the right side

x - 3 = \frac{3 π}{2} + 2 πn - 1

Add

3

to both sides

x - 3 + 3 = \frac{3 π}{2} + 2 πn - 1 + 3

Simplify

x - 3 + 3 = \frac{3 π}{2} + 2 πn - 1 + 3

Simplify

x - 3 + 3 : x

x - 3 + 3

Add similar elements:

- 3 + 3 = 0

= x

Simplify

\frac{3 π}{2} + 2 πn - 1 + 3 : 2 πn + 2 + \frac{3 π}{2}

\frac{3 π}{2} + 2 πn - 1 + 3

Add/Subtract the numbers:

- 1 + 3 = 2

= 2 πn + 2 + \frac{3 π}{2}

x = 2 πn + 2 + \frac{3 π}{2}

x = 2 πn + 2 + \frac{3 π}{2}

x = 2 πn + 2 + \frac{3 π}{2}

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

Popular Examples

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\frac{4 sec ^{2} ( x )}{2} = - 8 sec (x)

tan(x)+tan^3(x)=0

tan (x) + tan^{3} (x) = 0

sin^3(x)-2sin(x)=4cos^2(x)-3

sin^{3} (x) - 2 sin (x) = 4 cos^{2} (x) - 3

cos^2(x)+6sin(x)-6=0

cos^{2} (x) + 6 sin (x) - 6 = 0

(1-sin(x))*cos^3(x)=0

(1 - sin (x)) \cdot cos^{3} (x) = 0

Frequently Asked Questions (FAQ)

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