What is the general solution for cos^3(x)+sin^2(x)*cos(x)=-1/2 ?

The general solution for cos^3(x)+sin^2(x)*cos(x)=-1/2 is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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cos^3(x)+sin^2(x)*cos(x)=-1/2

Solution

cos^{3} (x) + sin^{2} (x) \cdot cos (x) = - \frac{1}{2}

Solution

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

cos^{3} (x) + sin^{2} (x) cos (x) = - \frac{1}{2}

Subtract

- \frac{1}{2}

from both sides

cos^{3} (x) + sin^{2} (x) cos (x) + \frac{1}{2} = 0

Simplify

cos^{3} (x) + sin^{2} (x) cos (x) + \frac{1}{2} : \frac{2 cos ^{3} ( x ) + 2 sin ^{2} ( x ) cos ( x ) + 1}{2}

cos^{3} (x) + sin^{2} (x) cos (x) + \frac{1}{2}

Convert element to fraction:

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

= \frac{cos ^{3} ( x ) \cdot 2}{2} + \frac{sin ^{2} ( x ) cos ( x ) \cdot 2}{2} + \frac{1}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ^{3} ( x ) \cdot 2 + sin ^{2} ( x ) cos ( x ) \cdot 2 + 1}{2}

\frac{2 cos ^{3} ( x ) + 2 sin ^{2} ( x ) cos ( x ) + 1}{2} = 0

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

2 cos^{3} (x) + 2 sin^{2} (x) cos (x) + 1 = 0

Rewrite using trig identities

1 + 2 cos^{3} (x) + 2 cos (x) sin^{2} (x)

Use the Pythagorean identity:

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

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

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

Simplify

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

1 + 2 cos^{3} (x) + 2 cos (x) (1 - cos^{2} (x))

Expand

2 cos (x) (1 - cos^{2} (x)) : 2 cos (x) - 2 cos^{3} (x)

2 cos (x) (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 2 cos (x), b = 1, c = cos^{2} (x)

= 2 cos (x) \cdot 1 - 2 cos (x) cos^{2} (x)

= 2 \cdot 1 \cdot cos (x) - 2 cos^{2} (x) cos (x)

Simplify

2 \cdot 1 \cdot cos (x) - 2 cos^{2} (x) cos (x) : 2 cos (x) - 2 cos^{3} (x)

2 \cdot 1 \cdot cos (x) - 2 cos^{2} (x) cos (x)

2 \cdot 1 \cdot cos (x) = 2 cos (x)

2 \cdot 1 \cdot cos (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos (x)

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

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

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (x)

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

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

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

Simplify

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

1 + 2 cos^{3} (x) + 2 cos (x) - 2 cos^{3} (x)

Group like terms

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

Add similar elements:

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

= 2 cos (x) + 1

= 2 cos (x) + 1

= 2 cos (x) + 1

1 + 2 cos (x) = 0

Move

1

to the right side

1 + 2 cos (x) = 0

Subtract

1

from both sides

1 + 2 cos (x) - 1 = 0 - 1

Simplify

2 cos (x) = - 1

2 cos (x) = - 1

Divide both sides by

2

2 cos (x) = - 1

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{- 1}{2}

Simplify

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2}

General solutions for

cos (x) = - \frac{1}{2}

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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Popular Examples

cos(3A)+sin(A)=0

cos (3 A) + sin (A) = 0

0=1-sec^2(x)

0 = 1 - sec^{2} (x)

sin(θ)=0.1

sin (θ) = 0.1

2/(sin(x))+10=6

\frac{2}{sin ( x )} + 10 = 6

sin(θ)-0.2cos(θ)=0.6377

sin (θ) - 0.2 cos (θ) = 0.6377

Frequently Asked Questions (FAQ)

What is the general solution for cos^3(x)+sin^2(x)*cos(x)=-1/2 ?
The general solution for cos^3(x)+sin^2(x)*cos(x)=-1/2 is x=(2pi)/3+2pin,x=(4pi)/3+2pin