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Popular >

5cos^2(θ)+cos(θ)=0

Solution

5 cos^{2} (θ) + cos (θ) = 0

Solution

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 1.77215 \dots + 2 πn, θ = - 1.77215 \dots + 2 πn

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n, θ = 101.53695 \dots^{\circ} + 36 0^{\circ} n, θ = - 101.53695 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 cos^{2} (θ) + cos (θ) = 0

Solve by substitution

5 cos^{2} (θ) + cos (θ) = 0

Let:

cos (θ) = u

5 u^{2} + u = 0

5 u^{2} + u = 0 : u = 0, u = - \frac{1}{5}

5 u^{2} + u = 0

Solve with the quadratic formula

5 u^{2} + u = 0

Quadratic Equation Formula:

For

a = 5, b = 1, c = 0

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 5 \cdot 0}{2 \cdot 5}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 5 \cdot 0}{2 \cdot 5}

1^{2} - 4 \cdot 5 \cdot 0 = 1

1^{2} - 4 \cdot 5 \cdot 0

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 5 \cdot 0

Apply rule

0 \cdot a = 0

= 1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- 1 \pm 1}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- 1 + 1}{2 \cdot 5}, u_{2} = \frac{- 1 - 1}{2 \cdot 5}

u = \frac{- 1 + 1}{2 \cdot 5} : 0

\frac{- 1 + 1}{2 \cdot 5}

Add/Subtract the numbers:

- 1 + 1 = 0

= \frac{0}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{0}{10}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

u = \frac{- 1 - 1}{2 \cdot 5} : - \frac{1}{5}

\frac{- 1 - 1}{2 \cdot 5}

Subtract the numbers:

- 1 - 1 = - 2

= \frac{- 2}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 2}{10}

Apply the fraction rule:

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

= - \frac{2}{10}

Cancel the common factor:

2

= - \frac{1}{5}

The solutions to the quadratic equation are:

u = 0, u = - \frac{1}{5}

Substitute back

u = cos (θ)

cos (θ) = 0, cos (θ) = - \frac{1}{5}

cos (θ) = 0, cos (θ) = - \frac{1}{5}

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

cos (θ) = 0

General solutions for

cos (θ) = 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}

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

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

cos (θ) = - \frac{1}{5} : θ = arccos (- \frac{1}{5}) + 2 πn, θ = - arccos (- \frac{1}{5}) + 2 πn

cos (θ) = - \frac{1}{5}

Apply trig inverse properties

cos (θ) = - \frac{1}{5}

General solutions for

cos (θ) = - \frac{1}{5}

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

θ = arccos (- \frac{1}{5}) + 2 πn, θ = - arccos (- \frac{1}{5}) + 2 πn

θ = arccos (- \frac{1}{5}) + 2 πn, θ = - arccos (- \frac{1}{5}) + 2 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = arccos (- \frac{1}{5}) + 2 πn, θ = - arccos (- \frac{1}{5}) + 2 πn

Show solutions in decimal form

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 1.77215 \dots + 2 πn, θ = - 1.77215 \dots + 2 πn

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