What is the general solution for csc(x+10)=3 ?

The general solution for csc(x+10)=3 is x=0.33983…+360n-10,x=180-0.33983…+360n-10

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csc(x+10)=3

csc (x + 1 0^{\circ}) = 3

x = 0.33983 \dots + 36 0^{\circ} n - 1 0^{\circ}, x = 18 0^{\circ} - 0.33983 \dots + 36 0^{\circ} n - 1 0^{\circ}

Radians

x = 0.33983 \dots - \frac{π}{18} + 2 πn, x = π - 0.33983 \dots - \frac{π}{18} + 2 πn

Solution steps

csc (x + 1 0^{\circ}) = 3

Apply trig inverse properties

csc (x + 1 0^{\circ}) = 3

General solutions for

csc (x + 1 0^{\circ}) = 3

csc (x) = a \Rightarrow x = arccsc (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arccsc (a) + 36 0^{\circ} n

x + 1 0^{\circ} = arccsc (3) + 36 0^{\circ} n, x + 1 0^{\circ} = 18 0^{\circ} - arccsc (3) + 36 0^{\circ} n

x + 1 0^{\circ} = arccsc (3) + 36 0^{\circ} n, x + 1 0^{\circ} = 18 0^{\circ} - arccsc (3) + 36 0^{\circ} n

Solve

x + 1 0^{\circ} = arccsc (3) + 36 0^{\circ} n : x = arccsc (3) + 36 0^{\circ} n - 1 0^{\circ}

x + 1 0^{\circ} = arccsc (3) + 36 0^{\circ} n

Move

1 0^{\circ}

to the right side

x + 1 0^{\circ} = arccsc (3) + 36 0^{\circ} n

Subtract

1 0^{\circ}

from both sides

x + 1 0^{\circ} - 1 0^{\circ} = arccsc (3) + 36 0^{\circ} n - 1 0^{\circ}

Simplify

x = arccsc (3) + 36 0^{\circ} n - 1 0^{\circ}

x = arccsc (3) + 36 0^{\circ} n - 1 0^{\circ}