What is the general solution for sin(x+20)*csc(70)=1 ?

The general solution for sin(x+20)*csc(70)=1 is x=1.22173…+360n-20,x=180-1.22173…+360n-20

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sin(x+20)*csc(70)=1

sin (x + 2 0^{\circ}) \cdot csc (7 0^{\circ}) = 1

x = 1.22173 \dots + 36 0^{\circ} n - 2 0^{\circ}, x = 18 0^{\circ} - 1.22173 \dots + 36 0^{\circ} n - 2 0^{\circ}

Radians

x = 1.22173 \dots - \frac{π}{9} + 2 πn, x = π - 1.22173 \dots - \frac{π}{9} + 2 πn

Solution steps

sin (x + 2 0^{\circ}) csc (7 0^{\circ}) = 1

Divide both sides by

csc (7 0^{\circ})

sin (x + 2 0^{\circ}) csc (7 0^{\circ}) = 1

Divide both sides by

csc (7 0^{\circ})

\frac{sin ( x + 2 0 ^{\circ} ) csc ( 7 0 ^{\circ} )}{csc ( 7 0 ^{\circ} )} = \frac{1}{csc ( 7 0 ^{\circ} )}

Simplify

sin (x + 2 0^{\circ}) = \frac{1}{csc ( 7 0 ^{\circ} )}

sin (x + 2 0^{\circ}) = \frac{1}{csc ( 7 0 ^{\circ} )}

Apply trig inverse properties

sin (x + 2 0^{\circ}) = \frac{1}{csc ( 7 0 ^{\circ} )}

General solutions for

sin (x + 2 0^{\circ}) = \frac{1}{csc ( 7 0 ^{\circ} )}

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

x + 2 0^{\circ} = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n, x + 2 0^{\circ} = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n

x + 2 0^{\circ} = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n, x + 2 0^{\circ} = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n

Solve

x + 2 0^{\circ} = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n : x = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}

x + 2 0^{\circ} = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n

Move

2 0^{\circ}

to the right side

x + 2 0^{\circ} = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n

Subtract

2 0^{\circ}

from both sides

x + 2 0^{\circ} - 2 0^{\circ} = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}

Simplify

x = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}

x = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}

Solve

x + 2 0^{\circ} = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n : x = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}

x + 2 0^{\circ} = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n

Move

2 0^{\circ}

to the right side

x + 2 0^{\circ} = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n

Subtract

2 0^{\circ}

from both sides

x + 2 0^{\circ} - 2 0^{\circ} = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}

Simplify

x = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}

x = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}

x = arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}, x = 18 0^{\circ} - arcsin (\frac{1}{csc ( 7 0 ^{\circ} )}) + 36 0^{\circ} n - 2 0^{\circ}

Show solutions in decimal form

x = 1.22173 \dots + 36 0^{\circ} n - 2 0^{\circ}, x = 18 0^{\circ} - 1.22173 \dots + 36 0^{\circ} n - 2 0^{\circ}

\frac{π}{2} = \frac{arcsin ( x )}{c}

6 sin^{2} (x) + 16 sin (x) cos (x) - 6 cos^{2} (x) = 0

sec^{2} (t) = 0

x, t = tan (\frac{x}{2})

sin^{2} (x) = 8 cos^{2} (x)

What is the general solution for sin(x+20)*csc(70)=1 ?
The general solution for sin(x+20)*csc(70)=1 is x=1.22173…+360n-20,x=180-1.22173…+360n-20