What is the general solution for tan(3x+26)=cot(5x) ?

The general solution for tan(3x+26)=cot(5x) is x=8+(180n)/4 ,x=30.5+(180n)/4

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Popular Trigonometry >

tan(3x+26)=cot(5x)

Solution

tan (3 x + 2 6^{\circ}) = cot (5 x)

Solution

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}, x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

Radians

x = \frac{2 π}{45} + \frac{π}{4} n, x = \frac{61 π}{360} + \frac{π}{4} n

Solution steps

tan (3 x + 2 6^{\circ}) = cot (5 x)

Subtract

cot (5 x)

from both sides

tan (3 x + 2 6^{\circ}) - cot (5 x) = 0

Express with sin, cos

- cot (5 x) + tan (2 6^{\circ} + 3 x)

Use the basic trigonometric identity:

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

= - \frac{cos ( 5 x )}{sin ( 5 x )} + tan (2 6^{\circ} + 3 x)

Use the basic trigonometric identity:

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

= - \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

Simplify

- \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )} : \frac{- cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} ) + sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

- \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

\frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )} = \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )}

\frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

Join

2 6^{\circ} + 3 x : \frac{234 0 ^{\circ} + 270 x}{90}

2 6^{\circ} + 3 x

Convert element to fraction:

3 x = \frac{3 x 90}{90}

= 2 6^{\circ} + \frac{3 x \cdot 90}{90}

Since the denominators are equal, combine the fractions:

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

= \frac{234 0 ^{\circ} + 3 x \cdot 90}{90}

Multiply the numbers:

3 \cdot 90 = 270

= \frac{234 0 ^{\circ} + 270 x}{90}

= \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )}

Join

2 6^{\circ} + 3 x : \frac{234 0 ^{\circ} + 270 x}{90}

2 6^{\circ} + 3 x

Convert element to fraction:

3 x = \frac{3 x 90}{90}

= 2 6^{\circ} + \frac{3 x \cdot 90}{90}

Since the denominators are equal, combine the fractions:

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

= \frac{234 0 ^{\circ} + 3 x \cdot 90}{90}

Multiply the numbers:

3 \cdot 90 = 270

= \frac{234 0 ^{\circ} + 270 x}{90}

= \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )}

= - \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( \frac{270 x + 234 0 ^{\circ}}{90} )}{cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

Least Common Multiplier of

sin (5 x), cos (\frac{234 0 ^{\circ} + 270 x}{90}) : sin (5 x) cos (\frac{270 x + 234 0 ^{\circ}}{90})

sin (5 x), cos (\frac{234 0 ^{\circ} + 270 x}{90})

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (5 x)

cos (\frac{234 0 ^{\circ} + 270 x}{90})

= sin (5 x) cos (\frac{270 x + 234 0 ^{\circ}}{90})

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 (5 x) cos (\frac{270 x + 234 0 ^{\circ}}{90})

For

\frac{cos ( 5 x )}{sin ( 5 x )} :

multiply the denominator and numerator by

cos (\frac{270 x + 234 0 ^{\circ}}{90})

\frac{cos ( 5 x )}{sin ( 5 x )} = \frac{cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

For

\frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )} :

multiply the denominator and numerator by

sin (5 x)

\frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )} = \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}

= - \frac{cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )} + \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} ) + sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

= \frac{- cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} ) + sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

\frac{- cos ( 5 x ) cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) + sin ( 5 x ) sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )} = 0

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

- cos (5 x) cos (\frac{234 0 ^{\circ} + 270 x}{90}) + sin (5 x) sin (\frac{234 0 ^{\circ} + 270 x}{90}) = 0

Rewrite using trig identities

- cos (5 x) cos (\frac{234 0 ^{\circ} + 270 x}{90}) + sin (5 x) sin (\frac{234 0 ^{\circ} + 270 x}{90})

Use the Angle Sum identity:

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

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

= - cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90})

- cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

Divide both sides by

- 1

- cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

Divide both sides by

- 1

\frac{- cos ( 5 x + \frac{234 0 ^{\circ} + 270 x}{90} )}{- 1} = \frac{0}{- 1}

Simplify

cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

General solutions for

cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n, 5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n, 5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

Solve

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n : x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

90

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

90

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 9 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplify

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 9 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplify

5 x \cdot 90 : 450 x

5 x \cdot 90

Multiply the numbers:

5 \cdot 90 = 450

= 450 x

Simplify

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 : 234 0^{\circ} + 270 x

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90

Multiply fractions:

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

= \frac{( 234 0 ^{\circ} + 270 x ) \cdot 90}{90}

Cancel the common factor:

90

= 234 0^{\circ} + 270 x

Simplify

9 0^{\circ} \cdot 90 : 810 0^{\circ}

9 0^{\circ} \cdot 90

Multiply fractions:

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

= 810 0^{\circ}

Divide the numbers:

\frac{90}{2} = 45

= 810 0^{\circ}

Simplify

36 0^{\circ} n \cdot 90 : 3240 0^{\circ} n

36 0^{\circ} n \cdot 90

Multiply the numbers:

2 \cdot 90 = 180

= 3240 0^{\circ} n

450 x + 234 0^{\circ} + 270 x = 810 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

Move

234 0^{\circ}

to the right side

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

Subtract

234 0^{\circ}

from both sides

720 x + 234 0^{\circ} - 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n - 234 0^{\circ}

Simplify

720 x = 576 0^{\circ} + 3240 0^{\circ} n

720 x = 576 0^{\circ} + 3240 0^{\circ} n

Divide both sides by

720

720 x = 576 0^{\circ} + 3240 0^{\circ} n

Divide both sides by

720

\frac{720 x}{720} = 8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Simplify

\frac{720 x}{720} = 8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Simplify

\frac{720 x}{720} : x

\frac{720 x}{720}

Divide the numbers:

\frac{720}{720} = 1

= x

Simplify

8^{\circ} + \frac{3240 0 ^{\circ} n}{720} : 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Cancel

8^{\circ} : 8^{\circ}

8^{\circ}

Cancel the common factor:

16

= 8^{\circ}

= 8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Cancel

\frac{3240 0 ^{\circ} n}{720} : \frac{18 0 ^{\circ} n}{4}

\frac{3240 0 ^{\circ} n}{720}

Cancel the common factor:

180

= \frac{18 0 ^{\circ} n}{4}

= 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

Solve

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n : x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

90

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

90

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 27 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplify

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 27 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplify

5 x \cdot 90 : 450 x

5 x \cdot 90

Multiply the numbers:

5 \cdot 90 = 450

= 450 x

Simplify

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 : 234 0^{\circ} + 270 x

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90

Multiply fractions:

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

= \frac{( 234 0 ^{\circ} + 270 x ) \cdot 90}{90}

Cancel the common factor:

90

= 234 0^{\circ} + 270 x

Simplify

27 0^{\circ} \cdot 90 : 2430 0^{\circ}

27 0^{\circ} \cdot 90

Multiply fractions:

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

= 2430 0^{\circ}

Multiply the numbers:

3 \cdot 90 = 270

= 2430 0^{\circ}

Divide the numbers:

\frac{270}{2} = 135

= 2430 0^{\circ}

Simplify

36 0^{\circ} n \cdot 90 : 3240 0^{\circ} n

36 0^{\circ} n \cdot 90

Multiply the numbers:

2 \cdot 90 = 180

= 3240 0^{\circ} n

450 x + 234 0^{\circ} + 270 x = 2430 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

Move

234 0^{\circ}

to the right side

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

Subtract

234 0^{\circ}

from both sides

720 x + 234 0^{\circ} - 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n - 234 0^{\circ}

Simplify

720 x = 2196 0^{\circ} + 3240 0^{\circ} n

720 x = 2196 0^{\circ} + 3240 0^{\circ} n

Divide both sides by

720

720 x = 2196 0^{\circ} + 3240 0^{\circ} n

Divide both sides by

720

\frac{720 x}{720} = 30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Simplify

\frac{720 x}{720} = 30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Simplify

\frac{720 x}{720} : x

\frac{720 x}{720}

Divide the numbers:

\frac{720}{720} = 1

= x

Simplify

30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720} : 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Cancel

30. 5^{\circ} : 30. 5^{\circ}

30. 5^{\circ}

Cancel the common factor:

2

= 30. 5^{\circ}

= 30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Cancel

\frac{3240 0 ^{\circ} n}{720} : \frac{18 0 ^{\circ} n}{4}

\frac{3240 0 ^{\circ} n}{720}

Cancel the common factor:

180

= \frac{18 0 ^{\circ} n}{4}

= 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}, x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan(3x+26)=cot(5x) ?
The general solution for tan(3x+26)=cot(5x) is x=8+(180n)/4 ,x=30.5+(180n)/4