What is the general solution for 2tan(x)=tan(x)tan(x) ?

The general solution for 2tan(x)=tan(x)tan(x) is x=pin,x=1.10714…+pin

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Popular Trigonometry >

2tan(x)=tan(x)tan(x)

Solution

2 tan (x) = tan (x) tan (x)

Solution

x = πn, x = 1.10714 \dots + πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 tan (x) = tan (x) tan (x)

Solve by substitution

2 tan (x) = tan (x) tan (x)

Let:

tan (x) = u

2 u = uu

2 u = uu : u = 0, u = 2

2 u = uu

Move

uu

to the left side

2 u = uu

Subtract

uu

from both sides

2 u - uu = uu - uu

Simplify

2 u - uu = 0

2 u - uu = 0

Factor

2 u - uu : u (2 - u)

2 u - uu

Factor out common term

u

= u (2 - u)

u (2 - u) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 2 - u = 0

Solve

2 - u = 0 : u = 2

2 - u = 0

Move

2

to the right side

2 - u = 0

Subtract

2

from both sides

2 - u - 2 = 0 - 2

Simplify

- u = - 2

- u = - 2

Divide both sides by

- 1

- u = - 2

Divide both sides by

- 1

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

Simplify

u = 2

u = 2

The solutions to the quadratic equation are:

u = 0, u = 2

Substitute back

u = tan (x)

tan (x) = 0, tan (x) = 2

tan (x) = 0, tan (x) = 2

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) = 2 : x = arctan (2) + πn

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

Combine all the solutions

x = πn, x = arctan (2) + πn

Show solutions in decimal form

x = πn, x = 1.10714 \dots + πn

Graph

Popular Examples

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solvefor x,-ysin(x)=0

Frequently Asked Questions (FAQ)

What is the general solution for 2tan(x)=tan(x)tan(x) ?
The general solution for 2tan(x)=tan(x)tan(x) is x=pin,x=1.10714…+pin