What is the general solution for tan(t)= 3/(tan(t)) ?

The general solution for tan(t)= 3/(tan(t)) is t= pi/3+pin,t=(2pi)/3+pin

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Popular >

tan(t)= 3/(tan(t))

Solution

tan (t) = \frac{3}{tan ( t )}

Solution

t = \frac{π}{3} + πn, t = \frac{2 π}{3} + πn

Degrees

t = 6 0^{\circ} + 18 0^{\circ} n, t = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

tan (t) = \frac{3}{tan ( t )}

Solve by substitution

tan (t) = \frac{3}{tan ( t )}

Let:

tan (t) = u

u = \frac{3}{u}

u = \frac{3}{u} : u = 3, u = - 3

u = \frac{3}{u}

Multiply both sides by

u

u = \frac{3}{u}

Multiply both sides by

u

uu = \frac{3}{u} u

Simplify

uu : u^{2}

uu = \frac{3}{u} u

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

u^{2} = 3

u^{2} = 3

Solve

u^{2} = 3 : u = 3, u = - 3

u^{2} = 3

For

x^{2} = f (a)

the solutions are

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

u = 3, u = - 3

u = 3, u = - 3

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{3}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 3, u = - 3

Substitute back

u = tan (t)

tan (t) = 3, tan (t) = - 3

tan (t) = 3, tan (t) = - 3

tan (t) = 3 : t = \frac{π}{3} + πn

tan (t) = 3

General solutions for

tan (t) = 3

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}

t = \frac{π}{3} + πn

t = \frac{π}{3} + πn

tan (t) = - 3 : t = \frac{2 π}{3} + πn

tan (t) = - 3

General solutions for

tan (t) = - 3

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}

t = \frac{2 π}{3} + πn

t = \frac{2 π}{3} + πn

Combine all the solutions

t = \frac{π}{3} + πn, t = \frac{2 π}{3} + πn

Popular Examples

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tan (- x) = tan (2 x + 1)

(2cos(x)-sin(x))(1+sin(x))=cos^2(x)

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

sin(6x)cos(9x)-cos(6x)sin(9x)=-0.4

sin (6 x) cos (9 x) - cos (6 x) sin (9 x) = - 0.4

sin^2(x)=2tan^2(x)

sin^{2} (x) = 2 tan^{2} (x)

cos^3(x)=cos(2x+x)

cos^{3} (x) = cos (2 x + x)

Frequently Asked Questions (FAQ)

What is the general solution for tan(t)= 3/(tan(t)) ?
The general solution for tan(t)= 3/(tan(t)) is t= pi/3+pin,t=(2pi)/3+pin