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Popular Trigonometry >

cot(11x+1)=tan(6x+4)

Solution

cot (11 x + 1) = tan (6 x + 4)

Solution

x = - \frac{5}{17} + \frac{π}{34} + \frac{2 πn}{17}, x = - \frac{5}{17} + \frac{3 π}{34} + \frac{2 πn}{17}

Degrees

x = - 11.55758 \dots^{\circ} + 21.17647 \dots^{\circ} n, x = - 0.96934 \dots^{\circ} + 21.17647 \dots^{\circ} n

Solution steps

cot (11 x + 1) = tan (6 x + 4)

Subtract

tan (6 x + 4)

from both sides

cot (11 x + 1) - tan (6 x + 4) = 0

Express with sin, cos

cot (1 + 11 x) - tan (4 + 6 x)

Use the basic trigonometric identity:

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

= \frac{cos ( 1 + 11 x )}{sin ( 1 + 11 x )} - tan (4 + 6 x)

Use the basic trigonometric identity:

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

= \frac{cos ( 1 + 11 x )}{sin ( 1 + 11 x )} - \frac{sin ( 4 + 6 x )}{cos ( 4 + 6 x )}

Simplify

\frac{cos ( 1 + 11 x )}{sin ( 1 + 11 x )} - \frac{sin ( 4 + 6 x )}{cos ( 4 + 6 x )} : \frac{cos ( 1 + 11 x ) cos ( 6 x + 4 ) - sin ( 4 + 6 x ) sin ( 11 x + 1 )}{sin ( 11 x + 1 ) cos ( 6 x + 4 )}

\frac{cos ( 1 + 11 x )}{sin ( 1 + 11 x )} - \frac{sin ( 4 + 6 x )}{cos ( 4 + 6 x )}

Least Common Multiplier of

sin (1 + 11 x), cos (4 + 6 x) : sin (11 x + 1) cos (6 x + 4)

sin (1 + 11 x), cos (4 + 6 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (1 + 11 x)

cos (4 + 6 x)

= sin (11 x + 1) cos (6 x + 4)

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 (11 x + 1) cos (6 x + 4)

For

\frac{cos ( 1 + 11 x )}{sin ( 1 + 11 x )} :

multiply the denominator and numerator by

cos (6 x + 4)

\frac{cos ( 1 + 11 x )}{sin ( 1 + 11 x )} = \frac{cos ( 1 + 11 x ) cos ( 6 x + 4 )}{sin ( 1 + 11 x ) cos ( 6 x + 4 )}

For

\frac{sin ( 4 + 6 x )}{cos ( 4 + 6 x )} :

multiply the denominator and numerator by

sin (11 x + 1)

\frac{sin ( 4 + 6 x )}{cos ( 4 + 6 x )} = \frac{sin ( 4 + 6 x ) sin ( 11 x + 1 )}{cos ( 4 + 6 x ) sin ( 11 x + 1 )}

= \frac{cos ( 1 + 11 x ) cos ( 6 x + 4 )}{sin ( 1 + 11 x ) cos ( 6 x + 4 )} - \frac{sin ( 4 + 6 x ) sin ( 11 x + 1 )}{cos ( 4 + 6 x ) sin ( 11 x + 1 )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( 1 + 11 x ) cos ( 6 x + 4 ) - sin ( 4 + 6 x ) sin ( 11 x + 1 )}{sin ( 11 x + 1 ) cos ( 6 x + 4 )}

= \frac{cos ( 1 + 11 x ) cos ( 6 x + 4 ) - sin ( 4 + 6 x ) sin ( 11 x + 1 )}{sin ( 11 x + 1 ) cos ( 6 x + 4 )}

\frac{cos ( 1 + 11 x ) cos ( 4 + 6 x ) - sin ( 1 + 11 x ) sin ( 4 + 6 x )}{cos ( 4 + 6 x ) sin ( 1 + 11 x )} = 0

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

cos (1 + 11 x) cos (4 + 6 x) - sin (1 + 11 x) sin (4 + 6 x) = 0

Rewrite using trig identities

cos (1 + 11 x) cos (4 + 6 x) - sin (1 + 11 x) sin (4 + 6 x)

Use the Angle Sum identity:

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

= cos (1 + 11 x + 4 + 6 x)

cos (1 + 11 x + 4 + 6 x) = 0

General solutions for

cos (1 + 11 x + 4 + 6 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 + 11 x + 4 + 6 x = \frac{π}{2} + 2 πn, 1 + 11 x + 4 + 6 x = \frac{3 π}{2} + 2 πn

1 + 11 x + 4 + 6 x = \frac{π}{2} + 2 πn, 1 + 11 x + 4 + 6 x = \frac{3 π}{2} + 2 πn

Solve

1 + 11 x + 4 + 6 x = \frac{π}{2} + 2 πn : x = - \frac{5}{17} + \frac{π}{34} + \frac{2 πn}{17}

1 + 11 x + 4 + 6 x = \frac{π}{2} + 2 πn

Group like terms

11 x + 6 x + 1 + 4 = \frac{π}{2} + 2 πn

Add similar elements:

11 x + 6 x = 17 x

17 x + 1 + 4 = \frac{π}{2} + 2 πn

Add the numbers:

1 + 4 = 5

17 x + 5 = \frac{π}{2} + 2 πn

Move

5

to the right side

17 x + 5 = \frac{π}{2} + 2 πn

Subtract

5

from both sides

17 x + 5 - 5 = \frac{π}{2} + 2 πn - 5

Simplify

17 x = \frac{π}{2} + 2 πn - 5

17 x = \frac{π}{2} + 2 πn - 5

Divide both sides by

17

17 x = \frac{π}{2} + 2 πn - 5

Divide both sides by

17

\frac{17 x}{17} = \frac{\frac{π}{2}}{17} + \frac{2 πn}{17} - \frac{5}{17}

Simplify

\frac{17 x}{17} = \frac{\frac{π}{2}}{17} + \frac{2 πn}{17} - \frac{5}{17}

Simplify

\frac{17 x}{17} : x

\frac{17 x}{17}

Divide the numbers:

\frac{17}{17} = 1

= x

Simplify

\frac{\frac{π}{2}}{17} + \frac{2 πn}{17} - \frac{5}{17} : - \frac{5}{17} + \frac{π}{34} + \frac{2 πn}{17}

\frac{\frac{π}{2}}{17} + \frac{2 πn}{17} - \frac{5}{17}

Group like terms

= - \frac{5}{17} + \frac{2 πn}{17} + \frac{\frac{π}{2}}{17}

\frac{\frac{π}{2}}{17} = \frac{π}{34}

\frac{\frac{π}{2}}{17}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 17}

Multiply the numbers:

2 \cdot 17 = 34

= \frac{π}{34}

= - \frac{5}{17} + \frac{2 πn}{17} + \frac{π}{34}

Group like terms

= - \frac{5}{17} + \frac{π}{34} + \frac{2 πn}{17}

x = - \frac{5}{17} + \frac{π}{34} + \frac{2 πn}{17}

x = - \frac{5}{17} + \frac{π}{34} + \frac{2 πn}{17}

x = - \frac{5}{17} + \frac{π}{34} + \frac{2 πn}{17}

Solve

1 + 11 x + 4 + 6 x = \frac{3 π}{2} + 2 πn : x = - \frac{5}{17} + \frac{3 π}{34} + \frac{2 πn}{17}

1 + 11 x + 4 + 6 x = \frac{3 π}{2} + 2 πn

Group like terms

11 x + 6 x + 1 + 4 = \frac{3 π}{2} + 2 πn

Add similar elements:

11 x + 6 x = 17 x

17 x + 1 + 4 = \frac{3 π}{2} + 2 πn

Add the numbers:

1 + 4 = 5

17 x + 5 = \frac{3 π}{2} + 2 πn

Move

5

to the right side

17 x + 5 = \frac{3 π}{2} + 2 πn

Subtract

5

from both sides

17 x + 5 - 5 = \frac{3 π}{2} + 2 πn - 5

Simplify

17 x = \frac{3 π}{2} + 2 πn - 5

17 x = \frac{3 π}{2} + 2 πn - 5

Divide both sides by

17

17 x = \frac{3 π}{2} + 2 πn - 5

Divide both sides by

17

\frac{17 x}{17} = \frac{\frac{3 π}{2}}{17} + \frac{2 πn}{17} - \frac{5}{17}

Simplify

\frac{17 x}{17} = \frac{\frac{3 π}{2}}{17} + \frac{2 πn}{17} - \frac{5}{17}

Simplify

\frac{17 x}{17} : x

\frac{17 x}{17}

Divide the numbers:

\frac{17}{17} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{17} + \frac{2 πn}{17} - \frac{5}{17} : - \frac{5}{17} + \frac{3 π}{34} + \frac{2 πn}{17}

\frac{\frac{3 π}{2}}{17} + \frac{2 πn}{17} - \frac{5}{17}

Group like terms

= - \frac{5}{17} + \frac{2 πn}{17} + \frac{\frac{3 π}{2}}{17}

\frac{\frac{3 π}{2}}{17} = \frac{3 π}{34}

\frac{\frac{3 π}{2}}{17}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 17}

Multiply the numbers:

2 \cdot 17 = 34

= \frac{3 π}{34}

= - \frac{5}{17} + \frac{2 πn}{17} + \frac{3 π}{34}

Group like terms

= - \frac{5}{17} + \frac{3 π}{34} + \frac{2 πn}{17}

x = - \frac{5}{17} + \frac{3 π}{34} + \frac{2 πn}{17}

x = - \frac{5}{17} + \frac{3 π}{34} + \frac{2 πn}{17}

x = - \frac{5}{17} + \frac{3 π}{34} + \frac{2 πn}{17}

x = - \frac{5}{17} + \frac{π}{34} + \frac{2 πn}{17}, x = - \frac{5}{17} + \frac{3 π}{34} + \frac{2 πn}{17}

Graph

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