English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

tan(x)=(5+cos(x))/(6sin(x)cos(x))

Solution

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

Solution

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

Subtract

\frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

from both sides

tan (x) - \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )} = 0

Simplify

tan (x) - \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )} : \frac{6 tan ( x ) sin ( x ) cos ( x ) - 5 - cos ( x )}{6 sin ( x ) cos ( x )}

tan (x) - \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

Convert element to fraction:

tan (x) = \frac{tan ( x ) 6 sin ( x ) cos ( x )}{6 sin ( x ) cos ( x )}

= \frac{tan ( x ) \cdot 6 sin ( x ) cos ( x )}{6 sin ( x ) cos ( x )} - \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{tan ( x ) \cdot 6 sin ( x ) cos ( x ) - ( 5 + cos ( x ) )}{6 sin ( x ) cos ( x )}

Expand

tan (x) \cdot 6 sin (x) cos (x) - (5 + cos (x)) : tan (x) \cdot 6 sin (x) cos (x) - 5 - cos (x)

tan (x) \cdot 6 sin (x) cos (x) - (5 + cos (x))

= 6 tan (x) sin (x) cos (x) - (5 + cos (x))

- (5 + cos (x)) : - 5 - cos (x)

- (5 + cos (x))

Distribute parentheses

= - (5) - (cos (x))

Apply minus-plus rules

+ (- a) = - a

= - 5 - cos (x)

= tan (x) \cdot 6 sin (x) cos (x) - 5 - cos (x)

= \frac{6 tan ( x ) sin ( x ) cos ( x ) - 5 - cos ( x )}{6 sin ( x ) cos ( x )}

\frac{6 tan ( x ) sin ( x ) cos ( x ) - 5 - cos ( x )}{6 sin ( x ) cos ( x )} = 0

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

6 tan (x) sin (x) cos (x) - 5 - cos (x) = 0

Express with sin, cos

6 \cdot \frac{sin ( x )}{cos ( x )} sin (x) cos (x) - 5 - cos (x) = 0

Simplify

6 \cdot \frac{sin ( x )}{cos ( x )} sin (x) cos (x) - 5 - cos (x) : 6 sin^{2} (x) - 5 - cos (x)

6 \cdot \frac{sin ( x )}{cos ( x )} sin (x) cos (x) - 5 - cos (x)

6 \cdot \frac{sin ( x )}{cos ( x )} sin (x) cos (x) = 6 sin^{2} (x)

6 \cdot \frac{sin ( x )}{cos ( x )} sin (x) cos (x)

Multiply fractions:

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

= \frac{sin ( x ) \cdot 6 sin ( x ) cos ( x )}{cos ( x )}

Cancel the common factor:

cos (x)

= sin (x) \cdot 6 sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= 6 sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= 6 sin^{2} (x)

= 6 sin^{2} (x) - 5 - cos (x)

6 sin^{2} (x) - 5 - cos (x) = 0

Add

cos (x)

to both sides

6 sin^{2} (x) - 5 = cos (x)

Square both sides

(6 sin^{2} (x) - 5)^{2} = cos^{2} (x)

Subtract

cos^{2} (x)

from both sides

(6 sin^{2} (x) - 5)^{2} - cos^{2} (x) = 0

Factor

(6 sin^{2} (x) - 5)^{2} - cos^{2} (x) : (6 sin^{2} (x) - 5 + cos (x)) (6 sin^{2} (x) - 5 - cos (x))

(6 sin^{2} (x) - 5)^{2} - cos^{2} (x)

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(6 sin^{2} (x) - 5)^{2} - cos^{2} (x) = ((6 sin^{2} (x) - 5) + cos (x)) ((6 sin^{2} (x) - 5) - cos (x))

= ((6 sin^{2} (x) - 5) + cos (x)) ((6 sin^{2} (x) - 5) - cos (x))

Refine

= (6 sin^{2} (x) + cos (x) - 5) (6 sin^{2} (x) - cos (x) - 5)

(6 sin^{2} (x) - 5 + cos (x)) (6 sin^{2} (x) - 5 - cos (x)) = 0

Solving each part separately

6 sin^{2} (x) - 5 + cos (x) = 0 or 6 sin^{2} (x) - 5 - cos (x) = 0

6 sin^{2} (x) - 5 + cos (x) = 0 : x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

6 sin^{2} (x) - 5 + cos (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 5 + cos (x) + 6 (1 - cos^{2} (x))

Simplify

- 5 + cos (x) + 6 (1 - cos^{2} (x)) : cos (x) - 6 cos^{2} (x) + 1

- 5 + cos (x) + 6 (1 - cos^{2} (x))

Expand

6 (1 - cos^{2} (x)) : 6 - 6 cos^{2} (x)

6 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 6, b = 1, c = cos^{2} (x)

= 6 \cdot 1 - 6 cos^{2} (x)

Multiply the numbers:

6 \cdot 1 = 6

= 6 - 6 cos^{2} (x)

= - 5 + cos (x) + 6 - 6 cos^{2} (x)

Simplify

- 5 + cos (x) + 6 - 6 cos^{2} (x) : cos (x) - 6 cos^{2} (x) + 1

- 5 + cos (x) + 6 - 6 cos^{2} (x)

Group like terms

= cos (x) - 6 cos^{2} (x) - 5 + 6

Add/Subtract the numbers:

- 5 + 6 = 1

= cos (x) - 6 cos^{2} (x) + 1

= cos (x) - 6 cos^{2} (x) + 1

= cos (x) - 6 cos^{2} (x) + 1

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

Solve by substitution

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

Let:

cos (x) = u

1 + u - 6 u^{2} = 0

1 + u - 6 u^{2} = 0 : u = - \frac{1}{3}, u = \frac{1}{2}

1 + u - 6 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

- 6 u^{2} + u + 1 = 0

Solve with the quadratic formula

- 6 u^{2} + u + 1 = 0

Quadratic Equation Formula:

For

a = - 6, b = 1, c = 1

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 6 ) \cdot 1}{2 ( - 6 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 6 ) \cdot 1}{2 ( - 6 )}

1^{2} - 4 (- 6) \cdot 1 = 5

1^{2} - 4 (- 6) \cdot 1

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 6) \cdot 1

Apply rule

- (- a) = a

= 1 + 4 \cdot 6 \cdot 1

Multiply the numbers:

4 \cdot 6 \cdot 1 = 24

= 1 + 24

Add the numbers:

1 + 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 1 \pm 5}{2 ( - 6 )}

Separate the solutions

u_{1} = \frac{- 1 + 5}{2 ( - 6 )}, u_{2} = \frac{- 1 - 5}{2 ( - 6 )}

u = \frac{- 1 + 5}{2 ( - 6 )} : - \frac{1}{3}

\frac{- 1 + 5}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 + 5}{- 2 \cdot 6}

Add/Subtract the numbers:

- 1 + 5 = 4

= \frac{4}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{4}{- 12}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{4}{12}

Cancel the common factor:

4

= - \frac{1}{3}

u = \frac{- 1 - 5}{2 ( - 6 )} : \frac{1}{2}

\frac{- 1 - 5}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 5}{- 2 \cdot 6}

Subtract the numbers:

- 1 - 5 = - 6

= \frac{- 6}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 6}{- 12}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{6}{12}

Cancel the common factor:

6

= \frac{1}{2}

The solutions to the quadratic equation are:

u = - \frac{1}{3}, u = \frac{1}{2}

Substitute back

u = cos (x)

cos (x) = - \frac{1}{3}, cos (x) = \frac{1}{2}

cos (x) = - \frac{1}{3}, cos (x) = \frac{1}{2}

cos (x) = - \frac{1}{3} : x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn

cos (x) = - \frac{1}{3}

Apply trig inverse properties

cos (x) = - \frac{1}{3}

General solutions for

cos (x) = - \frac{1}{3}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{2}

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}

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

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

Combine all the solutions

x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

6 sin^{2} (x) - 5 - cos (x) = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

6 sin^{2} (x) - 5 - cos (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 5 - cos (x) + 6 (1 - cos^{2} (x))

Simplify

- 5 - cos (x) + 6 (1 - cos^{2} (x)) : - 6 cos^{2} (x) - cos (x) + 1

- 5 - cos (x) + 6 (1 - cos^{2} (x))

Expand

6 (1 - cos^{2} (x)) : 6 - 6 cos^{2} (x)

6 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 6, b = 1, c = cos^{2} (x)

= 6 \cdot 1 - 6 cos^{2} (x)

Multiply the numbers:

6 \cdot 1 = 6

= 6 - 6 cos^{2} (x)

= - 5 - cos (x) + 6 - 6 cos^{2} (x)

Simplify

- 5 - cos (x) + 6 - 6 cos^{2} (x) : - 6 cos^{2} (x) - cos (x) + 1

- 5 - cos (x) + 6 - 6 cos^{2} (x)

Group like terms

= - cos (x) - 6 cos^{2} (x) - 5 + 6

Add/Subtract the numbers:

- 5 + 6 = 1

= - 6 cos^{2} (x) - cos (x) + 1

= - 6 cos^{2} (x) - cos (x) + 1

= - 6 cos^{2} (x) - cos (x) + 1

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

Solve by substitution

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

Let:

cos (x) = u

1 - u - 6 u^{2} = 0

1 - u - 6 u^{2} = 0 : u = - \frac{1}{2}, u = \frac{1}{3}

1 - u - 6 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

- 6 u^{2} - u + 1 = 0

Solve with the quadratic formula

- 6 u^{2} - u + 1 = 0

Quadratic Equation Formula:

For

a = - 6, b = - 1, c = 1

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 6 ) \cdot 1}{2 ( - 6 )}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 6 ) \cdot 1}{2 ( - 6 )}

(- 1)^{2} - 4 (- 6) \cdot 1 = 5

(- 1)^{2} - 4 (- 6) \cdot 1

Apply rule

- (- a) = a

= (- 1)^{2} + 4 \cdot 6 \cdot 1

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 6 \cdot 1 = 24

4 \cdot 6 \cdot 1

Multiply the numbers:

4 \cdot 6 \cdot 1 = 24

= 24

= 1 + 24

Add the numbers:

1 + 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 1 ) \pm 5}{2 ( - 6 )}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 5}{2 ( - 6 )}, u_{2} = \frac{- ( - 1 ) - 5}{2 ( - 6 )}

u = \frac{- ( - 1 ) + 5}{2 ( - 6 )} : - \frac{1}{2}

\frac{- ( - 1 ) + 5}{2 ( - 6 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{1 + 5}{- 2 \cdot 6}

Add the numbers:

1 + 5 = 6

= \frac{6}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{6}{- 12}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{6}{12}

Cancel the common factor:

6

= - \frac{1}{2}

u = \frac{- ( - 1 ) - 5}{2 ( - 6 )} : \frac{1}{3}

\frac{- ( - 1 ) - 5}{2 ( - 6 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{1 - 5}{- 2 \cdot 6}

Subtract the numbers:

1 - 5 = - 4

= \frac{- 4}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 4}{- 12}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{12}

Cancel the common factor:

4

= \frac{1}{3}

The solutions to the quadratic equation are:

u = - \frac{1}{2}, u = \frac{1}{3}

Substitute back

u = cos (x)

cos (x) = - \frac{1}{2}, cos (x) = \frac{1}{3}

cos (x) = - \frac{1}{2}, cos (x) = \frac{1}{3}

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{1}{2}

General solutions for

cos (x) = - \frac{1}{2}

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}

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

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

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

cos (x) = \frac{1}{3}

Apply trig inverse properties

cos (x) = \frac{1}{3}

General solutions for

cos (x) = \frac{1}{3}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Combine all the solutions

x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

Remove the ones that don't agree with the equation.

Check the solution

arccos (- \frac{1}{3}) + 2 πn :

False

arccos (- \frac{1}{3}) + 2 πn

Plug in

n = 1

arccos (- \frac{1}{3}) + 2 π 1

For

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

plug in

x = arccos (- \frac{1}{3}) + 2 π 1

tan (arccos (- \frac{1}{3}) + 2 π 1) = \frac{5 + cos ( arccos ( - \frac{1}{3} ) + 2 π 1 )}{6 sin ( arccos ( - \frac{1}{3} ) + 2 π 1 ) cos ( arccos ( - \frac{1}{3} ) + 2 π 1 )}

Refine

- 2.82842 \dots = - 2.47487 \dots

\Rightarrow False

Check the solution

- arccos (- \frac{1}{3}) + 2 πn :

False

- arccos (- \frac{1}{3}) + 2 πn

Plug in

n = 1

- arccos (- \frac{1}{3}) + 2 π 1

For

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

plug in

x = - arccos (- \frac{1}{3}) + 2 π 1

tan (- arccos (- \frac{1}{3}) + 2 π 1) = \frac{5 + cos ( - arccos ( - \frac{1}{3} ) + 2 π 1 )}{6 sin ( - arccos ( - \frac{1}{3} ) + 2 π 1 ) cos ( - arccos ( - \frac{1}{3} ) + 2 π 1 )}

Refine

2.82842 \dots = 2.47487 \dots

\Rightarrow False

Check the solution

\frac{π}{3} + 2 πn :

False

\frac{π}{3} + 2 πn

Plug in

n = 1

\frac{π}{3} + 2 π 1

For

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

plug in

x = \frac{π}{3} + 2 π 1

tan (\frac{π}{3} + 2 π 1) = \frac{5 + cos ( \frac{π}{3} + 2 π 1 )}{6 sin ( \frac{π}{3} + 2 π 1 ) cos ( \frac{π}{3} + 2 π 1 )}

Refine

1.73205 \dots = 2.11695 \dots

\Rightarrow False

Check the solution

\frac{5 π}{3} + 2 πn :

False

\frac{5 π}{3} + 2 πn

Plug in

n = 1

\frac{5 π}{3} + 2 π 1

For

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

plug in

x = \frac{5 π}{3} + 2 π 1

tan (\frac{5 π}{3} + 2 π 1) = \frac{5 + cos ( \frac{5 π}{3} + 2 π 1 )}{6 sin ( \frac{5 π}{3} + 2 π 1 ) cos ( \frac{5 π}{3} + 2 π 1 )}

Refine

- 1.73205 \dots = - 2.11695 \dots

\Rightarrow False

Check the solution

\frac{2 π}{3} + 2 πn :

True

\frac{2 π}{3} + 2 πn

Plug in

n = 1

\frac{2 π}{3} + 2 π 1

For

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

plug in

x = \frac{2 π}{3} + 2 π 1

tan (\frac{2 π}{3} + 2 π 1) = \frac{5 + cos ( \frac{2 π}{3} + 2 π 1 )}{6 sin ( \frac{2 π}{3} + 2 π 1 ) cos ( \frac{2 π}{3} + 2 π 1 )}

Refine

- 1.73205 \dots = - 1.73205 \dots

\Rightarrow True

Check the solution

\frac{4 π}{3} + 2 πn :

True

\frac{4 π}{3} + 2 πn

Plug in

n = 1

\frac{4 π}{3} + 2 π 1

For

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

plug in

x = \frac{4 π}{3} + 2 π 1

tan (\frac{4 π}{3} + 2 π 1) = \frac{5 + cos ( \frac{4 π}{3} + 2 π 1 )}{6 sin ( \frac{4 π}{3} + 2 π 1 ) cos ( \frac{4 π}{3} + 2 π 1 )}

Refine

1.73205 \dots = 1.73205 \dots

\Rightarrow True

Check the solution

arccos (\frac{1}{3}) + 2 πn :

True

arccos (\frac{1}{3}) + 2 πn

Plug in

n = 1

arccos (\frac{1}{3}) + 2 π 1

For

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

plug in

x = arccos (\frac{1}{3}) + 2 π 1

tan (arccos (\frac{1}{3}) + 2 π 1) = \frac{5 + cos ( arccos ( \frac{1}{3} ) + 2 π 1 )}{6 sin ( arccos ( \frac{1}{3} ) + 2 π 1 ) cos ( arccos ( \frac{1}{3} ) + 2 π 1 )}

Refine

2.82842 \dots = 2.82842 \dots

\Rightarrow True

Check the solution

2 π - arccos (\frac{1}{3}) + 2 πn :

True

2 π - arccos (\frac{1}{3}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{1}{3}) + 2 π 1

For

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

plug in

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

tan (2 π - arccos (\frac{1}{3}) + 2 π 1) = \frac{5 + cos ( 2 π - arccos ( \frac{1}{3} ) + 2 π 1 )}{6 sin ( 2 π - arccos ( \frac{1}{3} ) + 2 π 1 ) cos ( 2 π - arccos ( \frac{1}{3} ) + 2 π 1 )}

Refine

- 2.82842 \dots = - 2.82842 \dots

\Rightarrow True

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Show solutions in decimal form

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

Popular Examples

cos(x-1)= 1/2 ,0<= x<= 360

cos (x - 1) = \frac{1}{2}, 0^{\circ} \leq x \leq 36 0^{\circ}

cos(t)= 21/29

cos (t) = \frac{21}{29}

cos(x)=sin(x-pi/3)

cos (x) = sin (x - \frac{π}{3})

sin(x+pi/4)+sin(x+pi/4)=-1

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

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

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