What is the general solution for arcsin(x-2)= pi/6 ?

The general solution for arcsin(x-2)= pi/6 is x= 5/2

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arcsin(x-2)= pi/6

arcsin (x - 2) = \frac{π}{6}

x = \frac{5}{2}

Solution steps

arcsin (x - 2) = \frac{π}{6}

Apply trig inverse properties

arcsin (x - 2) = \frac{π}{6}

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

x - 2 = sin (\frac{π}{6})

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

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{1}{2}

= \frac{1}{2}

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

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

Solve

x - 2 = \frac{1}{2} : x = \frac{5}{2}

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

Move

2

to the right side

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

Add

2

to both sides

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

Simplify

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

Simplify

x - 2 + 2 : x

x - 2 + 2

Add similar elements:

- 2 + 2 = 0

= x

Simplify

\frac{1}{2} + 2 : \frac{5}{2}

\frac{1}{2} + 2

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

2 \cdot 2 + 1 = 5

2 \cdot 2 + 1

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 1

Add the numbers:

4 + 1 = 5

= 5

= \frac{5}{2}

x = \frac{5}{2}

x = \frac{5}{2}

x = \frac{5}{2}

x = \frac{5}{2}

What is the general solution for arcsin(x-2)= pi/6 ?
The general solution for arcsin(x-2)= pi/6 is x= 5/2