What is the general solution for sqrt(-9cos(θ))=(3sqrt(2))/2 ?

The general solution for sqrt(-9cos(θ))=(3sqrt(2))/2 is θ=(2pi)/3+2pin,θ=(4pi)/3+2pin

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sqrt(-9cos(θ))=(3sqrt(2))/2

Solution

- 9 cos (θ) = \frac{3 2}{2}

Solution

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

Solution steps

- 9 cos (θ) = \frac{3 2}{2}

Square both sides:

- 9 cos (θ) = \frac{9}{2}

- 9 cos (θ) = \frac{3 2}{2}

(- 9 cos (θ))^{2} = (\frac{3 2}{2})^{2}

Expand

(- 9 cos (θ))^{2} : - 9 cos (θ)

(- 9 cos (θ))^{2}

Apply radical rule:

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

= ((- 9 cos (θ))^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (- 9 cos (θ))^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= - 9 cos (θ)

Expand

(\frac{3 2}{2})^{2} : \frac{9}{2}

(\frac{3 2}{2})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 2 ) ^{2}}{2 ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(32)^{2} = 3^{2} (2)^{2}

= \frac{3 ^{2} ( 2 ) ^{2}}{2 ^{2}}

(2)^{2} : 2

Apply radical rule:

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

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

= \frac{3 ^{2} \cdot 2}{2 ^{2}}

Cancel the common factor:

2

= \frac{3 ^{2}}{2}

3^{2} = 9

= \frac{9}{2}

- 9 cos (θ) = \frac{9}{2}

- 9 cos (θ) = \frac{9}{2}

Solve

- 9 cos (θ) = \frac{9}{2} : cos (θ) = - \frac{1}{2}

- 9 cos (θ) = \frac{9}{2}

Divide both sides by

- 9

- 9 cos (θ) = \frac{9}{2}

Divide both sides by

- 9

\frac{- 9 cos ( θ )}{- 9} = \frac{\frac{9}{2}}{- 9}

Simplify

\frac{- 9 cos ( θ )}{- 9} = \frac{\frac{9}{2}}{- 9}

Simplify

\frac{- 9 cos ( θ )}{- 9} : cos (θ)

\frac{- 9 cos ( θ )}{- 9}

Apply the fraction rule:

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

= \frac{9 cos ( θ )}{9}

Divide the numbers:

\frac{9}{9} = 1

= cos (θ)

Simplify

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

\frac{\frac{9}{2}}{- 9}

Apply the fraction rule:

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

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

Apply the fraction rule:

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

\frac{\frac{9}{2}}{9} = \frac{9}{2 \cdot 9}

= - \frac{9}{2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= - \frac{9}{18}

Cancel the common factor:

9

= - \frac{1}{2}

cos (θ) = - \frac{1}{2}

cos (θ) = - \frac{1}{2}

cos (θ) = - \frac{1}{2}

cos (θ) = - \frac{1}{2}

Verify Solutions:

cos (θ) = - \frac{1}{2}

True

Check the solutions by plugging them into

- 9 cos (θ) = \frac{3 2}{2}

Remove the ones that don't agree with the equation.

Plug in

cos (θ) = - \frac{1}{2} :

True

- 9 (- \frac{1}{2}) = \frac{3 2}{2}

- 9 (- \frac{1}{2}) = \frac{3}{2}

- 9 (- \frac{1}{2})

Apply rule

- (- a) = a

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

Multiply

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

9 \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 9 = 9

= \frac{9}{2}

= \frac{9}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{2}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{2}

\frac{3 2}{2} = \frac{3}{2}

\frac{3 2}{2}

Apply radical rule:

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

= \frac{3 \cdot 2 ^{\frac{1}{2}}}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{3}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{3}{2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{3}{2}

\frac{3}{2} = \frac{3}{2}

True

The solution is

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

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

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Frequently Asked Questions (FAQ)

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