What is the general solution for cos(θ)=(-3+(-6))/((sqrt(9))(\sqrt{18))} ?

The general solution for cos(θ)=(-3+(-6))/((sqrt(9))(\sqrt{18))} is θ=(3pi)/4+2pin,θ=(5pi)/4+2pin

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cos(θ)=(-3+(-6))/((sqrt(9))(\sqrt{18))}

Solution

cos (θ) = \frac{- 3 + ( - 6 )}{( 9 ) ( 18 )}

Solution

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Solution steps

cos (θ) = \frac{- 3 + ( - 6 )}{( 9 ) ( 18 )}

Simplify

\frac{- 3 - 6}{9 18} : - \frac{2}{2}

\frac{- 3 - 6}{9 18}

Subtract the numbers:

- 3 - 6 = - 9

= \frac{- 9}{9 18}

Apply the fraction rule:

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

= - \frac{9}{9 18}

Cancel

\frac{9}{9 18} : \frac{1}{2}

\frac{9}{9 18}

Cancel

\frac{9}{9 18} : \frac{9}{18}

\frac{9}{9 18}

Apply radical rule:

9 = 9^{\frac{1}{2}}

= \frac{9}{9 ^{\frac{1}{2}} 18}

Apply exponent rule:

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

\frac{9 ^{1}}{9 ^{\frac{1}{2}}} = 9^{1 - \frac{1}{2}}

= \frac{9 ^{1 - \frac{1}{2}}}{18}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{9 ^{\frac{1}{2}}}{18}

Apply radical rule:

9^{\frac{1}{2}} = 9

= \frac{9}{18}

= \frac{9}{18}

Factor

9 : 3

Factor

9 = 3^{2}

= 3^{2}

Simplify

3^{2} : 3

3^{2}

Apply radical rule:

assuming

a \geq 0

= 3

= 3

Factor

18 : 32

Factor

18 = 3^{2} \cdot 2

= 3^{2} \cdot 2

Apply radical rule:

= 2 3^{2}

Simplify

3^{2} : 3

3^{2}

Apply radical rule:

assuming

a \geq 0

= 3

= 32

= \frac{3}{3 2}

Divide the numbers:

\frac{3}{3} = 1

= \frac{1}{2}

= - \frac{1}{2}

Rationalize

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

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

cos (θ) = - \frac{2}{2}

General solutions for

cos (θ) = - \frac{2}{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{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Popular Examples

sin(6/x)=36

sin (\frac{6}{x}) = 36

8cos(2x)+6=cos^2(x)+cos(x)

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

tan(x)= 200/500

tan (x) = \frac{200}{500}

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

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

sec(θ)=tan(θ)

sec (θ) = tan (θ)

Frequently Asked Questions (FAQ)

What is the general solution for cos(θ)=(-3+(-6))/((sqrt(9))(\sqrt{18))} ?
The general solution for cos(θ)=(-3+(-6))/((sqrt(9))(\sqrt{18))} is θ=(3pi)/4+2pin,θ=(5pi)/4+2pin