What is the general solution for 6arccos(3x)=5pi ?

The general solution for 6arccos(3x)=5pi is x=-1/(2sqrt(3))

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6arccos(3x)=5pi

Solution

6 arccos (3 x) = 5 π

Solution

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

Solution steps

6 arccos (3 x) = 5 π

Divide both sides by

6

6 arccos (3 x) = 5 π

Divide both sides by

6

\frac{6 arccos ( 3 x )}{6} = \frac{5 π}{6}

Simplify

arccos (3 x) = \frac{5 π}{6}

arccos (3 x) = \frac{5 π}{6}

Apply trig inverse properties

arccos (3 x) = \frac{5 π}{6}

arccos (x) = a \Rightarrow x = cos (a)

3 x = cos (\frac{5 π}{6})

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 π}{6})

Use the following trivial identity:

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 π}{6})

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}

= - \frac{3}{2}

= - \frac{3}{2}

3 x = - \frac{3}{2}

3 x = - \frac{3}{2}

Solve

3 x = - \frac{3}{2} : x = - \frac{1}{2 3}

3 x = - \frac{3}{2}

Divide both sides by

3

3 x = - \frac{3}{2}

Divide both sides by

3

\frac{3 x}{3} = \frac{- \frac{3}{2}}{3}

Simplify

\frac{3 x}{3} = \frac{- \frac{3}{2}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Cancel the common factor:

3

= x

Simplify

\frac{- \frac{3}{2}}{3} : - \frac{1}{2 3}

\frac{- \frac{3}{2}}{3}

Apply the fraction rule:

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

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

Apply the fraction rule:

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

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

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

Multiply the numbers:

2 \cdot 3 = 6

= - \frac{3}{6}

Cancel

- \frac{3}{6} : - \frac{1}{3 \cdot 2}

- \frac{3}{6}

Factor the number:

6 = 3 \cdot 2

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

Apply radical rule:

a = a a

3 = 33

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

Cancel the common factor:

3

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

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

= - \frac{1}{2 3}

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

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

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

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

Graph

Popular Examples

2sin(θ)=-sqrt(3),360,180<= θ

2 sin (θ) = - 3, 36 0^{\circ}, 18 0^{\circ} \leq θ

tan(θ)=(-3)/4

tan (θ) = \frac{- 3}{4}

sin(2x)cos(x)=cos(2x)sin(x)

sin (2 x) cos (x) = cos (2 x) sin (x)

sin(x)= 7/12

sin (x) = \frac{7}{12}

sin(x)= 7/13

sin (x) = \frac{7}{13}

Frequently Asked Questions (FAQ)

What is the general solution for 6arccos(3x)=5pi ?
The general solution for 6arccos(3x)=5pi is x=-1/(2sqrt(3))