What is the general solution for tan(x)=-24/7 tan(2x) ?

The general solution for tan(x)=-24/7 tan(2x) is x=pin,x=-1.22811…+pin,x=1.22811…+pin

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Popular Trigonometry >

tan(x)=-24/7 tan(2x)

Solution

tan (x) = - \frac{24}{7} tan (2 x)

Solution

x = πn, x = - 1.22811 \dots + πn, x = 1.22811 \dots + πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = - 70.36598 \dots^{\circ} + 18 0^{\circ} n, x = 70.36598 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan (x) = - \frac{24}{7} tan (2 x)

Subtract

- \frac{24}{7} tan (2 x)

from both sides

tan (x) + \frac{24}{7} tan (2 x) = 0

Simplify

tan (x) + \frac{24}{7} tan (2 x) : \frac{7 tan ( x ) + 24 tan ( 2 x )}{7}

tan (x) + \frac{24}{7} tan (2 x)

Multiply

\frac{24}{7} tan (2 x) : \frac{24 tan ( 2 x )}{7}

\frac{24}{7} tan (2 x)

Multiply fractions:

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

= \frac{24 tan ( 2 x )}{7}

= tan (x) + \frac{24 tan ( 2 x )}{7}

Convert element to fraction:

tan (x) = \frac{tan ( x ) 7}{7}

= \frac{tan ( x ) \cdot 7}{7} + \frac{24 tan ( 2 x )}{7}

Since the denominators are equal, combine the fractions:

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

= \frac{tan ( x ) \cdot 7 + 24 tan ( 2 x )}{7}

\frac{7 tan ( x ) + 24 tan ( 2 x )}{7} = 0

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

7 tan (x) + 24 tan (2 x) = 0

Rewrite using trig identities

24 tan (2 x) + 7 tan (x)

Use the Double Angle identity:

tan (2 x) = \frac{2 tan ( x )}{1 - tan ^{2} ( x )}

= 24 \cdot \frac{2 tan ( x )}{1 - tan ^{2} ( x )} + 7 tan (x)

Simplify

24 \cdot \frac{2 tan ( x )}{1 - tan ^{2} ( x )} + 7 tan (x) : \frac{55 tan ( x ) - 7 tan ^{3} ( x )}{1 - tan ^{2} ( x )}

24 \cdot \frac{2 tan ( x )}{1 - tan ^{2} ( x )} + 7 tan (x)

24 \cdot \frac{2 tan ( x )}{1 - tan ^{2} ( x )} = \frac{48 tan ( x )}{1 - tan ^{2} ( x )}

24 \cdot \frac{2 tan ( x )}{1 - tan ^{2} ( x )}

Multiply fractions:

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

= \frac{2 tan ( x ) \cdot 24}{1 - tan ^{2} ( x )}

Multiply the numbers:

2 \cdot 24 = 48

= \frac{48 tan ( x )}{1 - tan ^{2} ( x )}

= \frac{48 tan ( x )}{- tan ^{2} ( x ) + 1} + 7 tan (x)

Convert element to fraction:

7 tan (x) = \frac{7 tan ( x ) ( 1 - tan ^{2} ( x ) )}{1 - tan ^{2} ( x )}

= \frac{48 tan ( x )}{1 - tan ^{2} ( x )} + \frac{7 tan ( x ) ( 1 - tan ^{2} ( x ) )}{1 - tan ^{2} ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{48 tan ( x ) + 7 tan ( x ) ( 1 - tan ^{2} ( x ) )}{1 - tan ^{2} ( x )}

Expand

48 tan (x) + 7 tan (x) (1 - tan^{2} (x)) : 55 tan (x) - 7 tan^{3} (x)

48 tan (x) + 7 tan (x) (1 - tan^{2} (x))

Expand

7 tan (x) (1 - tan^{2} (x)) : 7 tan (x) - 7 tan^{3} (x)

7 tan (x) (1 - tan^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 7 tan (x), b = 1, c = tan^{2} (x)

= 7 tan (x) \cdot 1 - 7 tan (x) tan^{2} (x)

= 7 \cdot 1 \cdot tan (x) - 7 tan^{2} (x) tan (x)

Simplify

7 \cdot 1 \cdot tan (x) - 7 tan^{2} (x) tan (x) : 7 tan (x) - 7 tan^{3} (x)

7 \cdot 1 \cdot tan (x) - 7 tan^{2} (x) tan (x)

7 \cdot 1 \cdot tan (x) = 7 tan (x)

7 \cdot 1 \cdot tan (x)

Multiply the numbers:

7 \cdot 1 = 7

= 7 tan (x)

7 tan^{2} (x) tan (x) = 7 tan^{3} (x)

7 tan^{2} (x) tan (x)

Apply exponent rule:

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

tan^{2} (x) tan (x) = tan^{2 + 1} (x)

= 7 tan^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= 7 tan^{3} (x)

= 7 tan (x) - 7 tan^{3} (x)

= 7 tan (x) - 7 tan^{3} (x)

= 48 tan (x) + 7 tan (x) - 7 tan^{3} (x)

Add similar elements:

48 tan (x) + 7 tan (x) = 55 tan (x)

= 55 tan (x) - 7 tan^{3} (x)

= \frac{55 tan ( x ) - 7 tan ^{3} ( x )}{1 - tan ^{2} ( x )}

= \frac{55 tan ( x ) - 7 tan ^{3} ( x )}{1 - tan ^{2} ( x )}

\frac{55 tan ( x ) - 7 tan ^{3} ( x )}{1 - tan ^{2} ( x )} = 0

Solve by substitution

\frac{55 tan ( x ) - 7 tan ^{3} ( x )}{1 - tan ^{2} ( x )} = 0

Let:

tan (x) = u

\frac{55 u - 7 u ^{3}}{1 - u ^{2}} = 0

\frac{55 u - 7 u ^{3}}{1 - u ^{2}} = 0 : u = 0, u = - \frac{55}{7}, u = \frac{55}{7}

\frac{55 u - 7 u ^{3}}{1 - u ^{2}} = 0

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

55 u - 7 u^{3} = 0

Solve

55 u - 7 u^{3} = 0 : u = 0, u = - \frac{55}{7}, u = \frac{55}{7}

55 u - 7 u^{3} = 0

Factor

55 u - 7 u^{3} : - u (7 u + 55) (7 u - 55)

55 u - 7 u^{3}

Factor out common term

- u : - u (7 u^{2} - 55)

- 7 u^{3} + 55 u

Apply exponent rule:

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

u^{3} = u^{2} u

= - 7 u^{2} u + 55 u

Factor out common term

- u

= - u (7 u^{2} - 55)

= - u (7 u^{2} - 55)

Factor

7 u^{2} - 55 : (7 u + 55) (7 u - 55)

7 u^{2} - 55

Rewrite

7 u^{2} - 55

(7 u)^{2} - (55)^{2}

7 u^{2} - 55

Apply radical rule:

a = (a)^{2}

7 = (7)^{2}

= (7)^{2} u^{2} - 55

Apply radical rule:

a = (a)^{2}

55 = (55)^{2}

= (7)^{2} u^{2} - (55)^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(7)^{2} u^{2} = (7 u)^{2}

= (7 u)^{2} - (55)^{2}

= (7 u)^{2} - (55)^{2}

Apply Difference of Two Squares Formula:

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

(7 u)^{2} - (55)^{2} = (7 u + 55) (7 u - 55)

= (7 u + 55) (7 u - 55)

= - u (7 u + 55) (7 u - 55)

- u (7 u + 55) (7 u - 55) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 7 u + 55 = 0 or 7 u - 55 = 0

Solve

7 u + 55 = 0 : u = - \frac{55}{7}

7 u + 55 = 0

Move

55

to the right side

7 u + 55 = 0

Subtract

55

from both sides

7 u + 55 - 55 = 0 - 55

Simplify

7 u = - 55

7 u = - 55

Divide both sides by

7

7 u = - 55

Divide both sides by

7

\frac{7 u}{7} = \frac{- 55}{7}

Simplify

\frac{7 u}{7} = \frac{- 55}{7}

Simplify

\frac{7 u}{7} : u

\frac{7 u}{7}

Cancel the common factor:

7

= u

Simplify

\frac{- 55}{7} : - \frac{55}{7}

\frac{- 55}{7}

Apply the fraction rule:

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

= - \frac{55}{7}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{55}{7}

u = - \frac{55}{7}

u = - \frac{55}{7}

u = - \frac{55}{7}

Solve

7 u - 55 = 0 : u = \frac{55}{7}

7 u - 55 = 0

Move

55

to the right side

7 u - 55 = 0

Add

55

to both sides

7 u - 55 + 55 = 0 + 55

Simplify

7 u = 55

7 u = 55

Divide both sides by

7

7 u = 55

Divide both sides by

7

\frac{7 u}{7} = \frac{55}{7}

Simplify

\frac{7 u}{7} = \frac{55}{7}

Simplify

\frac{7 u}{7} : u

\frac{7 u}{7}

Cancel the common factor:

7

= u

Simplify

\frac{55}{7} : \frac{55}{7}

\frac{55}{7}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{55}{7}

u = \frac{55}{7}

u = \frac{55}{7}

u = \frac{55}{7}

The solutions are

u = 0, u = - \frac{55}{7}, u = \frac{55}{7}

u = 0, u = - \frac{55}{7}, u = \frac{55}{7}

Verify Solutions

Find undefined (singularity) points:

u = 1, u = - 1

Take the denominator(s) of

\frac{55 u - 7 u ^{3}}{1 - u ^{2}}

and compare to zero

Solve

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

Move

1

to the right side

1 - u^{2} = 0

Subtract

1

from both sides

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

Simplify

- u^{2} = - 1

- u^{2} = - 1

Divide both sides by

- 1

- u^{2} = - 1

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = \frac{- 1}{- 1}

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Apply radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

The following points are undefined

u = 1, u = - 1

Combine undefined points with solutions:

u = 0, u = - \frac{55}{7}, u = \frac{55}{7}

Substitute back

u = tan (x)

tan (x) = 0, tan (x) = - \frac{55}{7}, tan (x) = \frac{55}{7}

tan (x) = 0, tan (x) = - \frac{55}{7}, tan (x) = \frac{55}{7}

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

tan (x) = - \frac{55}{7} : x = arctan (- \frac{55}{7}) + πn

tan (x) = - \frac{55}{7}

Apply trig inverse properties

tan (x) = - \frac{55}{7}

General solutions for

tan (x) = - \frac{55}{7}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{55}{7}) + πn

x = arctan (- \frac{55}{7}) + πn

tan (x) = \frac{55}{7} : x = arctan (\frac{55}{7}) + πn

tan (x) = \frac{55}{7}

Apply trig inverse properties

tan (x) = \frac{55}{7}

General solutions for

tan (x) = \frac{55}{7}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{55}{7}) + πn

x = arctan (\frac{55}{7}) + πn

Combine all the solutions

x = πn, x = arctan (- \frac{55}{7}) + πn, x = arctan (\frac{55}{7}) + πn

Show solutions in decimal form

x = πn, x = - 1.22811 \dots + πn, x = 1.22811 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)=-24/7 tan(2x) ?
The general solution for tan(x)=-24/7 tan(2x) is x=pin,x=-1.22811…+pin,x=1.22811…+pin