What is the general solution for tan^2(x)sin(x)-(sin(x))/3 =0 ?

The general solution for tan^2(x)sin(x)-(sin(x))/3 =0 is x=2pin,x=pi+2pin,x=(5pi)/6+pin,x= pi/6+pin

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Popular Trigonometry >

tan^2(x)sin(x)-(sin(x))/3 =0

Solution

tan^{2} (x) sin (x) - \frac{sin ( x )}{3} = 0

Solution

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) sin (x) - \frac{sin ( x )}{3} = 0

Simplify

tan^{2} (x) sin (x) - \frac{sin ( x )}{3} : \frac{3 tan ^{2} ( x ) sin ( x ) - sin ( x )}{3}

tan^{2} (x) sin (x) - \frac{sin ( x )}{3}

Convert element to fraction:

tan^{2} (x) sin (x) = \frac{tan ^{2} ( x ) sin ( x ) 3}{3}

= \frac{tan ^{2} ( x ) sin ( x ) \cdot 3}{3} - \frac{sin ( x )}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{tan ^{2} ( x ) sin ( x ) \cdot 3 - sin ( x )}{3}

\frac{3 tan ^{2} ( x ) sin ( x ) - sin ( x )}{3} = 0

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

3 tan^{2} (x) sin (x) - sin (x) = 0

Factor

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

3 tan^{2} (x) sin (x) - sin (x)

Factor out common term

sin (x)

= sin (x) (3 tan^{2} (x) - 1)

Factor

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

3 tan^{2} (x) - 1

Rewrite

3 tan^{2} (x) - 1

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

3 tan^{2} (x) - 1

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} tan^{2} (x) - 1

Rewrite

1

1^{2}

= (3)^{2} tan^{2} (x) - 1^{2}

Apply exponent rule:

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

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

= (3 tan (x))^{2} - 1^{2}

= (3 tan (x))^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

(3 tan (x))^{2} - 1^{2} = (3 tan (x) + 1) (3 tan (x) - 1)

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

= sin (x) (3 tan (x) + 1) (3 tan (x) - 1)

sin (x) (3 tan (x) + 1) (3 tan (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 3 tan (x) + 1 = 0 or 3 tan (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

3 tan (x) + 1 = 0 : x = \frac{5 π}{6} + πn

3 tan (x) + 1 = 0

Move

1

to the right side

3 tan (x) + 1 = 0

Subtract

1

from both sides

3 tan (x) + 1 - 1 = 0 - 1

Simplify

3 tan (x) = - 1

3 tan (x) = - 1

Divide both sides by

3

3 tan (x) = - 1

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

Cancel the common factor:

3

= tan (x)

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

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

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

General solutions for

tan (x) = - \frac{3}{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}

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

3 tan (x) - 1 = 0 : x = \frac{π}{6} + πn

3 tan (x) - 1 = 0

Move

1

to the right side

3 tan (x) - 1 = 0

Add

1

to both sides

3 tan (x) - 1 + 1 = 0 + 1

Simplify

3 tan (x) = 1

3 tan (x) = 1

Divide both sides by

3

3 tan (x) = 1

Divide both sides by

3

\frac{3 tan ( x )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( x )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

Cancel the common factor:

3

= tan (x)

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

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

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

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

General solutions for

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

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)sin(x)-(sin(x))/3 =0 ?
The general solution for tan^2(x)sin(x)-(sin(x))/3 =0 is x=2pin,x=pi+2pin,x=(5pi)/6+pin,x= pi/6+pin