What is the general solution for 14tan^2(x)=-14tan(x) ?

The general solution for 14tan^2(x)=-14tan(x) is x=pin,x=(3pi)/4+pin

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Popular Trigonometry >

14tan^2(x)=-14tan(x)

Solution

14 tan^{2} (x) = - 14 tan (x)

Solution

x = πn, x = \frac{3 π}{4} + πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

14 tan^{2} (x) = - 14 tan (x)

Solve by substitution

14 tan^{2} (x) = - 14 tan (x)

Let:

tan (x) = u

14 u^{2} = - 14 u

14 u^{2} = - 14 u : u = 0, u = - 1

14 u^{2} = - 14 u

Move

14 u

to the left side

14 u^{2} = - 14 u

Add

14 u

to both sides

14 u^{2} + 14 u = - 14 u + 14 u

Simplify

14 u^{2} + 14 u = 0

14 u^{2} + 14 u = 0

Factor

14 u^{2} + 14 u : 14 u (u + 1)

14 u^{2} + 14 u

Apply exponent rule:

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

u^{2} = uu

= 14 uu + 14 u

Factor out common term

14 u

= 14 u (u + 1)

14 u (u + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or u + 1 = 0

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

The solutions to the quadratic equation are:

u = 0, u = - 1

Substitute back

u = tan (x)

tan (x) = 0, tan (x) = - 1

tan (x) = 0, tan (x) = - 1

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) = - 1 : x = \frac{3 π}{4} + πn

tan (x) = - 1

General solutions for

tan (x) = - 1

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 = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

x = πn, x = \frac{3 π}{4} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 14tan^2(x)=-14tan(x) ?
The general solution for 14tan^2(x)=-14tan(x) is x=pin,x=(3pi)/4+pin