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Popular >

6tan(x)(1-cos^2(x))=tan(x)

Solution

6 tan (x) (1 - cos^{2} (x)) = tan (x)

Solution

x = πn, x = 2.72105 \dots + 2 πn, x = - 2.72105 \dots + 2 πn, x = 0.42053 \dots + 2 πn, x = 2 π - 0.42053 \dots + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 155.90515 \dots^{\circ} + 36 0^{\circ} n, x = - 155.90515 \dots^{\circ} + 36 0^{\circ} n, x = 24.09484 \dots^{\circ} + 36 0^{\circ} n, x = 335.90515 \dots^{\circ} + 36 0^{\circ} n

Solution steps

6 tan (x) (1 - cos^{2} (x)) = tan (x)

Subtract

tan (x)

from both sides

5 tan (x) - 6 cos^{2} (x) tan (x) = 0

Factor

5 tan (x) - 6 cos^{2} (x) tan (x) : - tan (x) (6 cos (x) + 5) (6 cos (x) - 5)

5 tan (x) - 6 cos^{2} (x) tan (x)

Factor out common term

- tan (x)

= - tan (x) (- 5 + 6 cos^{2} (x))

Factor

6 cos^{2} (x) - 5 : (6 cos (x) + 5) (6 cos (x) - 5)

6 cos^{2} (x) - 5

Rewrite

6 cos^{2} (x) - 5

(6 cos (x))^{2} - (5)^{2}

6 cos^{2} (x) - 5

Apply radical rule:

a = (a)^{2}

6 = (6)^{2}

= (6)^{2} cos^{2} (x) - 5

Apply radical rule:

a = (a)^{2}

5 = (5)^{2}

= (6)^{2} cos^{2} (x) - (5)^{2}

Apply exponent rule:

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

(6)^{2} cos^{2} (x) = (6 cos (x))^{2}

= (6 cos (x))^{2} - (5)^{2}

= (6 cos (x))^{2} - (5)^{2}

Apply Difference of Two Squares Formula:

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

(6 cos (x))^{2} - (5)^{2} = (6 cos (x) + 5) (6 cos (x) - 5)

= (6 cos (x) + 5) (6 cos (x) - 5)

= - tan (x) (6 cos (x) + 5) (6 cos (x) - 5)

- tan (x) (6 cos (x) + 5) (6 cos (x) - 5) = 0

Solving each part separately

tan (x) = 0 or 6 cos (x) + 5 = 0 or 6 cos (x) - 5 = 0

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

6 cos (x) + 5 = 0 : x = arccos (- \frac{5}{6}) + 2 πn, x = - arccos (- \frac{5}{6}) + 2 πn

6 cos (x) + 5 = 0

Move

5

to the right side

6 cos (x) + 5 = 0

Subtract

5

from both sides

6 cos (x) + 5 - 5 = 0 - 5

Simplify

6 cos (x) = - 5

6 cos (x) = - 5

Divide both sides by

6

6 cos (x) = - 5

Divide both sides by

6

\frac{6 cos ( x )}{6} = \frac{- 5}{6}

Simplify

\frac{6 cos ( x )}{6} = \frac{- 5}{6}

Simplify

\frac{6 cos ( x )}{6} : cos (x)

\frac{6 cos ( x )}{6}

Cancel the common factor:

6

= cos (x)

Simplify

\frac{- 5}{6} : - \frac{5}{6}

\frac{- 5}{6}

Apply the fraction rule:

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

= - \frac{5}{6}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{5}{6}

cos (x) = - \frac{5}{6}

cos (x) = - \frac{5}{6}

cos (x) = - \frac{5}{6}

Apply trig inverse properties

cos (x) = - \frac{5}{6}

General solutions for

cos (x) = - \frac{5}{6}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{5}{6}) + 2 πn, x = - arccos (- \frac{5}{6}) + 2 πn

x = arccos (- \frac{5}{6}) + 2 πn, x = - arccos (- \frac{5}{6}) + 2 πn

6 cos (x) - 5 = 0 : x = arccos (\frac{5}{6}) + 2 πn, x = 2 π - arccos (\frac{5}{6}) + 2 πn

6 cos (x) - 5 = 0

Move

5

to the right side

6 cos (x) - 5 = 0

Add

5

to both sides

6 cos (x) - 5 + 5 = 0 + 5

Simplify

6 cos (x) = 5

6 cos (x) = 5

Divide both sides by

6

6 cos (x) = 5

Divide both sides by

6

\frac{6 cos ( x )}{6} = \frac{5}{6}

Simplify

\frac{6 cos ( x )}{6} = \frac{5}{6}

Simplify

\frac{6 cos ( x )}{6} : cos (x)

\frac{6 cos ( x )}{6}

Cancel the common factor:

6

= cos (x)

Simplify

\frac{5}{6} : \frac{5}{6}

\frac{5}{6}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{5}{6}

cos (x) = \frac{5}{6}

cos (x) = \frac{5}{6}

cos (x) = \frac{5}{6}

Apply trig inverse properties

cos (x) = \frac{5}{6}

General solutions for

cos (x) = \frac{5}{6}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{5}{6}) + 2 πn, x = 2 π - arccos (\frac{5}{6}) + 2 πn

x = arccos (\frac{5}{6}) + 2 πn, x = 2 π - arccos (\frac{5}{6}) + 2 πn

Combine all the solutions

x = πn, x = arccos (- \frac{5}{6}) + 2 πn, x = - arccos (- \frac{5}{6}) + 2 πn, x = arccos (\frac{5}{6}) + 2 πn, x = 2 π - arccos (\frac{5}{6}) + 2 πn

Show solutions in decimal form

x = πn, x = 2.72105 \dots + 2 πn, x = - 2.72105 \dots + 2 πn, x = 0.42053 \dots + 2 πn, x = 2 π - 0.42053 \dots + 2 πn

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