What is the general solution for (178.9)/(sin(125))=(125)/(sin(x)) ?

The general solution for (178.9)/(sin(125))=(125)/(sin(x)) is x=0.60937…+360n,x=180-0.60937…+360n

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(178.9)/(sin(125))=(125)/(sin(x))

\frac{178.9}{sin ( 12 5 ^{\circ} )} = \frac{125}{sin ( x )}

x = 0.60937 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.60937 \dots + 36 0^{\circ} n

Radians

x = 0.60937 \dots + 2 πn, x = π - 0.60937 \dots + 2 πn

Solution steps

\frac{178.9}{sin ( 12 5 ^{\circ} )} = \frac{125}{sin ( x )}

Cross multiply

\frac{178.9}{sin ( 12 5 ^{\circ} )} = \frac{125}{sin ( x )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

178.9 sin (x) = sin (12 5^{\circ}) \cdot 125

178.9 sin (x) = sin (12 5^{\circ}) \cdot 125

Divide both sides by

178.9

178.9 sin (x) = sin (12 5^{\circ}) \cdot 125

Divide both sides by

178.9

\frac{178.9 sin ( x )}{178.9} = \frac{sin ( 12 5 ^{\circ} ) \cdot 125}{178.9}

Simplify

sin (x) = \frac{sin ( 12 5 ^{\circ} ) \cdot 125}{178.9}

sin (x) = \frac{sin ( 12 5 ^{\circ} ) \cdot 125}{178.9}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{125}{sin ( x )}

and compare to zero

sin (x) = 0

The following points are undefined

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{sin ( 12 5 ^{\circ} ) \cdot 125}{178.9}

Apply trig inverse properties

sin (x) = \frac{sin ( 12 5 ^{\circ} ) \cdot 125}{178.9}

General solutions for

sin (x) = \frac{sin ( 12 5 ^{\circ} ) 125}{178.9}

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

x = arcsin (\frac{sin ( 12 5 ^{\circ} ) \cdot 125}{178.9}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{sin ( 12 5 ^{\circ} ) \cdot 125}{178.9}) + 36 0^{\circ} n

x = arcsin (\frac{sin ( 12 5 ^{\circ} ) \cdot 125}{178.9}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{sin ( 12 5 ^{\circ} ) \cdot 125}{178.9}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.60937 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.60937 \dots + 36 0^{\circ} n

cos (θ) = \frac{3}{8}

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

\frac{1}{4} sin (4 t) = 0

2 cos (4 x) = 2 cos^{2} (2 x) - sin (2 x) - 1

1 - sin^{2} (θ) = 0

What is the general solution for (178.9)/(sin(125))=(125)/(sin(x)) ?
The general solution for (178.9)/(sin(125))=(125)/(sin(x)) is x=0.60937…+360n,x=180-0.60937…+360n