What is the general solution for arccos((sqrt(3))/2)+arcsin(3x)= pi/2 ?

The general solution for arccos((sqrt(3))/2)+arcsin(3x)= pi/2 is x= 1/(2sqrt(3))

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arccos((sqrt(3))/2)+arcsin(3x)= pi/2

Solution

arccos (\frac{3}{2}) + arcsin (3 x) = \frac{π}{2}

Solution

x = \frac{1}{2 3}

Solution steps

arccos (\frac{3}{2}) + arcsin (3 x) = \frac{π}{2}

arccos (\frac{3}{2}) = \frac{π}{6}

arccos (\frac{3}{2})

Use the following trivial identity:

arccos (\frac{3}{2}) = \frac{π}{6}

arccos (\frac{3}{2})

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{6}

= \frac{π}{6}

\frac{π}{6} + arcsin (3 x) = \frac{π}{2}

Move

\frac{π}{6}

to the right side

\frac{π}{6} + arcsin (3 x) = \frac{π}{2}

Subtract

\frac{π}{6}

from both sides

\frac{π}{6} + arcsin (3 x) - \frac{π}{6} = \frac{π}{2} - \frac{π}{6}

Simplify

\frac{π}{6} + arcsin (3 x) - \frac{π}{6} = \frac{π}{2} - \frac{π}{6}

Simplify

\frac{π}{6} + arcsin (3 x) - \frac{π}{6} : arcsin (3 x)

\frac{π}{6} + arcsin (3 x) - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= arcsin (3 x)

Simplify

\frac{π}{2} - \frac{π}{6} : \frac{π}{3}

\frac{π}{2} - \frac{π}{6}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

6

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

= \frac{π 3}{6} - \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π}{6}

Add similar elements:

3 π - π = 2 π

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

arcsin (3 x) = \frac{π}{3}

arcsin (3 x) = \frac{π}{3}

arcsin (3 x) = \frac{π}{3}

Apply trig inverse properties

arcsin (3 x) = \frac{π}{3}

arcsin (x) = a \Rightarrow x = sin (a)

3 x = sin (\frac{π}{3})

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{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{\frac{a}{b}}{c} = \frac{a}{b \cdot c}

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

Cancel

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

\frac{3}{2 \cdot 3}

Apply radical rule:

a = a a

3 = 33

= \frac{3}{2 3 3}

Cancel the common factor:

3

= \frac{1}{2 3}

= \frac{1}{2 3}

x = \frac{1}{2 3}

x = \frac{1}{2 3}

x = \frac{1}{2 3}

x = \frac{1}{2 3}

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Frequently Asked Questions (FAQ)

What is the general solution for arccos((sqrt(3))/2)+arcsin(3x)= pi/2 ?
The general solution for arccos((sqrt(3))/2)+arcsin(3x)= pi/2 is x= 1/(2sqrt(3))