What is the general solution for -1=(tan(x)-(3/4))/(1+tan(x)(3/4)) ?

The general solution for -1=(tan(x)-(3/4))/(1+tan(x)(3/4)) is x=-0.14189…+pin

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Popular Trigonometry >

-1=(tan(x)-(3/4))/(1+tan(x)(3/4))

Solution

- 1 = \frac{tan ( x ) - ( \frac{3}{4} )}{1 + tan ( x ) ( \frac{3}{4} )}

Solution

x = - 0.14189 \dots + πn

Degrees

x = - 8.13010 \dots^{\circ} + 18 0^{\circ} n

Solution steps

- 1 = \frac{tan ( x ) - ( \frac{3}{4} )}{1 + tan ( x ) ( \frac{3}{4} )}

Switch sides

\frac{tan ( x ) - \frac{3}{4}}{1 + tan ( x ) \frac{3}{4}} = - 1

Solve by substitution

\frac{tan ( x ) - \frac{3}{4}}{1 + tan ( x ) \frac{3}{4}} = - 1

Let:

tan (x) = u

\frac{u - \frac{3}{4}}{1 + u \frac{3}{4}} = - 1

\frac{u - \frac{3}{4}}{1 + u \frac{3}{4}} = - 1 : u = - \frac{1}{7}

\frac{u - \frac{3}{4}}{1 + u \frac{3}{4}} = - 1

Simplify

\frac{u - \frac{3}{4}}{1 + u \frac{3}{4}} : \frac{4 u - 3}{4 + 3 u}

\frac{u - \frac{3}{4}}{1 + u \frac{3}{4}}

Join

1 + u \frac{3}{4} : \frac{4 + 3 u}{4}

1 + u \frac{3}{4}

Multiply

u \frac{3}{4} : \frac{3 u}{4}

u \frac{3}{4}

Multiply fractions:

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

= \frac{3 u}{4}

= 1 + \frac{3 u}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} + \frac{3 u}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 + 3 u}{4}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4 + 3 u}{4}

= \frac{u - \frac{3}{4}}{\frac{4 + 3 u}{4}}

Join

u - \frac{3}{4} : \frac{4 u - 3}{4}

u - \frac{3}{4}

Convert element to fraction:

u = \frac{u 4}{4}

= \frac{u \cdot 4}{4} - \frac{3}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{u \cdot 4 - 3}{4}

= \frac{\frac{4 u - 3}{4}}{\frac{4 + 3 u}{4}}

Divide fractions:

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

= \frac{( u \cdot 4 - 3 ) \cdot 4}{4 ( 4 + 3 u )}

Cancel the common factor:

4

= \frac{u \cdot 4 - 3}{4 + 3 u}

\frac{4 u - 3}{4 + 3 u} = - 1

Multiply both sides by

4 + 3 u

\frac{4 u - 3}{4 + 3 u} = - 1

Multiply both sides by

4 + 3 u

\frac{4 u - 3}{4 + 3 u} (4 + 3 u) = - 1 \cdot (4 + 3 u)

Simplify

\frac{4 u - 3}{4 + 3 u} (4 + 3 u) = - 1 \cdot (4 + 3 u)

Simplify

\frac{4 u - 3}{4 + 3 u} (4 + 3 u) : 4 u - 3

\frac{4 u - 3}{4 + 3 u} (4 + 3 u)

Multiply fractions:

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

= \frac{( 4 u - 3 ) ( 4 + 3 u )}{4 + 3 u}

Cancel the common factor:

4 + 3 u

= 4 u - 3

Simplify

- 1 \cdot (4 + 3 u) : - (4 + 3 u)

- 1 \cdot (4 + 3 u)

Multiply:

1 \cdot (4 + 3 u) = (4 + 3 u)

= - (3 u + 4)

4 u - 3 = - (4 + 3 u)

4 u - 3 = - (4 + 3 u)

4 u - 3 = - (4 + 3 u)

Expand

- (4 + 3 u) : - 4 - 3 u

- (4 + 3 u)

Distribute parentheses

= - (4) - (3 u)

Apply minus-plus rules

+ (- a) = - a

= - 4 - 3 u

4 u - 3 = - 4 - 3 u

Move

3

to the right side

4 u - 3 = - 4 - 3 u

Add

3

to both sides

4 u - 3 + 3 = - 4 - 3 u + 3

Simplify

4 u = - 3 u - 1

4 u = - 3 u - 1

Move

3 u

to the left side

4 u = - 3 u - 1

Add

3 u

to both sides

4 u + 3 u = - 3 u - 1 + 3 u

Simplify

7 u = - 1

7 u = - 1

Divide both sides by

7

7 u = - 1

Divide both sides by

7

\frac{7 u}{7} = \frac{- 1}{7}

Simplify

u = - \frac{1}{7}

u = - \frac{1}{7}

Verify Solutions

Find undefined (singularity) points:

u = - \frac{4}{3}

Take the denominator(s) of

\frac{u - \frac{3}{4}}{1 + u \frac{3}{4}}

and compare to zero

Solve

1 + u \frac{3}{4} = 0 : u = - \frac{4}{3}

1 + u \frac{3}{4} = 0

Move

1

to the right side

1 + u \frac{3}{4} = 0

Subtract

1

from both sides

1 + u \frac{3}{4} - 1 = 0 - 1

Simplify

u \frac{3}{4} = - 1

u \frac{3}{4} = - 1

Multiply both sides by

4

u \frac{3}{4} = - 1

Multiply both sides by

4

4 u \frac{3}{4} = 4 (- 1)

Simplify

3 u = - 4

3 u = - 4

Divide both sides by

3

3 u = - 4

Divide both sides by

3

\frac{3 u}{3} = \frac{- 4}{3}

Simplify

u = - \frac{4}{3}

u = - \frac{4}{3}

The following points are undefined

u = - \frac{4}{3}

Combine undefined points with solutions:

u = - \frac{1}{7}

Substitute back

u = tan (x)

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

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

tan (x) = - \frac{1}{7} : x = arctan (- \frac{1}{7}) + πn

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan (- \frac{1}{7}) + πn

x = arctan (- \frac{1}{7}) + πn

Combine all the solutions

x = arctan (- \frac{1}{7}) + πn

Show solutions in decimal form

x = - 0.14189 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -1=(tan(x)-(3/4))/(1+tan(x)(3/4)) ?
The general solution for -1=(tan(x)-(3/4))/(1+tan(x)(3/4)) is x=-0.14189…+pin