What is the general solution for (31.5)/(sin(57.22))=(24.4)/(sin(x)) ?

The general solution for (31.5)/(sin(57.22))=(24.4)/(sin(x)) is x=0.70923…+360n,x=180-0.70923…+360n

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(31.5)/(sin(57.22))=(24.4)/(sin(x))

\frac{31.5}{sin ( 57.2 2 ^{\circ} )} = \frac{24.4}{sin ( x )}

x = 0.70923 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.70923 \dots + 36 0^{\circ} n

Radians

x = 0.70923 \dots + 2 πn, x = π - 0.70923 \dots + 2 πn

Solution steps

\frac{31.5}{sin ( 57.2 2 ^{\circ} )} = \frac{24.4}{sin ( x )}

Cross multiply

\frac{31.5}{sin ( 57.2 2 ^{\circ} )} = \frac{24.4}{sin ( x )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

31.5 sin (x) = sin (57.2 2^{\circ}) \cdot 24.4

31.5 sin (x) = sin (57.2 2^{\circ}) \cdot 24.4

Divide both sides by

31.5

31.5 sin (x) = sin (57.2 2^{\circ}) \cdot 24.4

Divide both sides by

31.5

\frac{31.5 sin ( x )}{31.5} = \frac{sin ( 57.2 2 ^{\circ} ) \cdot 24.4}{31.5}

Simplify

sin (x) = \frac{sin ( 57.2 2 ^{\circ} ) \cdot 24.4}{31.5}

sin (x) = \frac{sin ( 57.2 2 ^{\circ} ) \cdot 24.4}{31.5}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{24.4}{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 ( 57.2 2 ^{\circ} ) \cdot 24.4}{31.5}

Apply trig inverse properties

sin (x) = \frac{sin ( 57.2 2 ^{\circ} ) \cdot 24.4}{31.5}

General solutions for

sin (x) = \frac{sin ( 57.2 2 ^{\circ} ) 24.4}{31.5}

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 ( 57.2 2 ^{\circ} ) \cdot 24.4}{31.5}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{sin ( 57.2 2 ^{\circ} ) \cdot 24.4}{31.5}) + 36 0^{\circ} n

x = arcsin (\frac{sin ( 57.2 2 ^{\circ} ) \cdot 24.4}{31.5}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{sin ( 57.2 2 ^{\circ} ) \cdot 24.4}{31.5}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.70923 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.70923 \dots + 36 0^{\circ} n

23 cos (θ - \frac{π}{3}) = 3, 0 \leq θ \leq 2 π

sin (6 x) = 1

3 sin (θ) = 2 cos (θ)

\frac{2}{2} = sin (x)

tan (θ) = \frac{12}{6}

What is the general solution for (31.5)/(sin(57.22))=(24.4)/(sin(x)) ?
The general solution for (31.5)/(sin(57.22))=(24.4)/(sin(x)) is x=0.70923…+360n,x=180-0.70923…+360n