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

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

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

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

x = \frac{1}{2}

Solution steps

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

arcsin (\frac{3}{2}) = \frac{π}{3}

arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

arcsin (\frac{3}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{3}

= \frac{π}{3}

arccos (x) + 2 \cdot \frac{π}{3} = π

Move

2 \frac{π}{3}

to the right side

arccos (x) + 2 \cdot \frac{π}{3} = π

Subtract

2 \frac{π}{3}

from both sides

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

Simplify

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

Simplify

arccos (x) + 2 \cdot \frac{π}{3} - 2 \cdot \frac{π}{3} : arccos (x)

arccos (x) + 2 \cdot \frac{π}{3} - 2 \cdot \frac{π}{3}

Add similar elements:

2 \frac{π}{3} - 2 \frac{π}{3} = 0

= arccos (x)

Simplify

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

π - 2 \cdot \frac{π}{3}

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= π - \frac{2 π}{3}

arccos (x) = π - \frac{2 π}{3}

arccos (x) = π - \frac{2 π}{3}

arccos (x) = π - \frac{2 π}{3}

Apply trig inverse properties

arccos (x) = π - \frac{2 π}{3}

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

x = cos (π - \frac{2 π}{3})

cos (π - \frac{2 π}{3}) = \frac{1}{2}

cos (π - \frac{2 π}{3})

Simplify:

π - \frac{2 π}{3} = \frac{π}{3}

π - \frac{2 π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= \frac{π 3}{3} - \frac{2 π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - 2 π}{3}

Add similar elements:

3 π - 2 π = π

= \frac{π}{3}

= cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

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{1}{2}

= \frac{1}{2}

x = \frac{1}{2}

x = \frac{1}{2}

3 - 3 sin (θ) = 2 cos^{2} (θ)

tan (a) = - 3

cos^{2} (θ) = sin^{2} (θ)

csc (t) = \frac{1}{4}

sin (θ) = 0.21

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