What is the general solution for -10cos(θ)=-2-6cos(θ) ?

The general solution for -10cos(θ)=-2-6cos(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin

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Popular >

-10cos(θ)=-2-6cos(θ)

Solution

- 10 cos (θ) = - 2 - 6 cos (θ)

Solution

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

- 10 cos (θ) = - 2 - 6 cos (θ)

Solve by substitution

- 10 cos (θ) = - 2 - 6 cos (θ)

Let:

cos (θ) = u

- 10 u = - 2 - 6 u

- 10 u = - 2 - 6 u : u = \frac{1}{2}

- 10 u = - 2 - 6 u

Move

6 u

to the left side

- 10 u = - 2 - 6 u

Add

6 u

to both sides

- 10 u + 6 u = - 2 - 6 u + 6 u

Simplify

- 4 u = - 2

- 4 u = - 2

Divide both sides by

- 4

- 4 u = - 2

Divide both sides by

- 4

\frac{- 4 u}{- 4} = \frac{- 2}{- 4}

Simplify

\frac{- 4 u}{- 4} = \frac{- 2}{- 4}

Simplify

\frac{- 4 u}{- 4} : u

\frac{- 4 u}{- 4}

Apply the fraction rule:

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

= \frac{4 u}{4}

Divide the numbers:

\frac{4}{4} = 1

= u

Simplify

\frac{- 2}{- 4} : \frac{1}{2}

\frac{- 2}{- 4}

Apply the fraction rule:

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

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u = \frac{1}{2}

u = \frac{1}{2}

u = \frac{1}{2}

Substitute back

u = cos (θ)

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

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

cos (θ) = \frac{1}{2}

General solutions for

cos (θ) = \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}

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Combine all the solutions

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Popular Examples

sin(a)=(((8))/((sqrt(153))))

sin (a) = (\frac{( 8 )}{( 153 )})

2csc^2(x)-1=0

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

-sec(θ)cot(θ)-sec(θ)=-2sec(θ)

- sec (θ) cot (θ) - sec (θ) = - 2 sec (θ)

sec^2(a)= 4/3

sec^{2} (a) = \frac{4}{3}

sin(5x-27)=cos(6x+13.6)

sin (5 x - 27) = cos (6 x + 13.6)

Frequently Asked Questions (FAQ)

What is the general solution for -10cos(θ)=-2-6cos(θ) ?
The general solution for -10cos(θ)=-2-6cos(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin