What is the general solution for (9.03)/(sin(40))=(10)/(sin(x)) ?

The general solution for (9.03)/(sin(40))=(10)/(sin(x)) is x=0.79210…+360n,x=180-0.79210…+360n

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(9.03)/(sin(40))=(10)/(sin(x))

\frac{9.03}{sin ( 4 0 ^{\circ} )} = \frac{10}{sin ( x )}

x = 0.79210 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.79210 \dots + 36 0^{\circ} n

Radians

x = 0.79210 \dots + 2 πn, x = π - 0.79210 \dots + 2 πn

Solution steps

\frac{9.03}{sin ( 4 0 ^{\circ} )} = \frac{10}{sin ( x )}

Cross multiply

\frac{9.03}{sin ( 4 0 ^{\circ} )} = \frac{10}{sin ( x )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

9.03 sin (x) = sin (4 0^{\circ}) \cdot 10

9.03 sin (x) = sin (4 0^{\circ}) \cdot 10

Divide both sides by

9.03

9.03 sin (x) = sin (4 0^{\circ}) \cdot 10

Divide both sides by

9.03

\frac{9.03 sin ( x )}{9.03} = \frac{sin ( 4 0 ^{\circ} ) \cdot 10}{9.03}

Simplify

sin (x) = \frac{sin ( 4 0 ^{\circ} ) \cdot 10}{9.03}

sin (x) = \frac{sin ( 4 0 ^{\circ} ) \cdot 10}{9.03}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{10}{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 ( 4 0 ^{\circ} ) \cdot 10}{9.03}

Apply trig inverse properties

sin (x) = \frac{sin ( 4 0 ^{\circ} ) \cdot 10}{9.03}

General solutions for

sin (x) = \frac{sin ( 4 0 ^{\circ} ) 10}{9.03}

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 ( 4 0 ^{\circ} ) \cdot 10}{9.03}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{sin ( 4 0 ^{\circ} ) \cdot 10}{9.03}) + 36 0^{\circ} n

x = arcsin (\frac{sin ( 4 0 ^{\circ} ) \cdot 10}{9.03}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{sin ( 4 0 ^{\circ} ) \cdot 10}{9.03}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.79210 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.79210 \dots + 36 0^{\circ} n

What is the general solution for (9.03)/(sin(40))=(10)/(sin(x)) ?
The general solution for (9.03)/(sin(40))=(10)/(sin(x)) is x=0.79210…+360n,x=180-0.79210…+360n