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Popular >

2cos^3(x)-3sin^2(x)cos(x)=0

Solution

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

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.68471 \dots + πn, x = 0.68471 \dots + πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 39.23152 \dots^{\circ} + 18 0^{\circ} n, x = 39.23152 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Factor

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

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

Apply exponent rule:

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

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

= 2 cos (x) cos^{2} (x) - 3 sin^{2} (x) cos (x)

Factor out common term

cos (x)

= cos (x) (2 cos^{2} (x) - 3 sin^{2} (x))

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} cos^{2} (x) - 3 sin^{2} (x)

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (2)^{2} cos^{2} (x) - (3)^{2} sin^{2} (x)

Apply exponent rule:

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

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

= (2 cos (x))^{2} - (3)^{2} sin^{2} (x)

Apply exponent rule:

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

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

= (2 cos (x))^{2} - (3 sin (x))^{2}

= (2 cos (x))^{2} - (3 sin (x))^{2}

Apply Difference of Two Squares Formula:

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

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

= (2 cos (x) + 3 sin (x)) (2 cos (x) - 3 sin (x))

= cos (x) (2 cos (x) + 3 sin (x)) (2 cos (x) - 3 sin (x))

cos (x) (2 cos (x) + 3 sin (x)) (2 cos (x) - 3 sin (x)) = 0

Solving each part separately

cos (x) = 0 or 2 cos (x) + 3 sin (x) = 0 or 2 cos (x) - 3 sin (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

2 cos (x) + 3 sin (x) = 0 : x = arctan (- \frac{2}{3}) + πn

2 cos (x) + 3 sin (x) = 0

Rewrite using trig identities

2 cos (x) + 3 sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{2 cos ( x ) + 3 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

2 + \frac{3 sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

2 + 3 tan (x) = 0

2 + 3 tan (x) = 0

Move

2

to the right side

2 + 3 tan (x) = 0

Subtract

2

from both sides

2 + 3 tan (x) - 2 = 0 - 2

Simplify

3 tan (x) = - 2

3 tan (x) = - 2

Divide both sides by

3

3 tan (x) = - 2

Divide both sides by

3

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

Simplify

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

Simplify

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

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

Cancel the common factor:

3

= tan (x)

Simplify

\frac{- 2}{3} : - \frac{2}{3}

\frac{- 2}{3}

Apply the fraction rule:

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

= - \frac{2}{3}

Combine same powers :

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

= - \frac{2}{3}

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan (- \frac{2}{3}) + πn

x = arctan (- \frac{2}{3}) + πn

2 cos (x) - 3 sin (x) = 0 : x = arctan (\frac{2}{3}) + πn

2 cos (x) - 3 sin (x) = 0

Rewrite using trig identities

2 cos (x) - 3 sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

2 - 3 tan (x) = 0

2 - 3 tan (x) = 0

Move

2

to the right side

2 - 3 tan (x) = 0

Subtract

2

from both sides

2 - 3 tan (x) - 2 = 0 - 2

Simplify

- 3 tan (x) = - 2

- 3 tan (x) = - 2

Divide both sides by

- 3

- 3 tan (x) = - 2

Divide both sides by

- 3

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

Simplify

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

3

= tan (x)

Simplify

\frac{- 2}{- 3} : \frac{2}{3}

\frac{- 2}{- 3}

Apply the fraction rule:

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

= \frac{2}{3}

Combine same powers :

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

= \frac{2}{3}

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan (\frac{2}{3}) + πn

x = arctan (\frac{2}{3}) + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arctan (- \frac{2}{3}) + πn, x = arctan (\frac{2}{3}) + πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.68471 \dots + πn, x = 0.68471 \dots + πn

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