What is the general solution for (16)/(sin(a))=(12)/(sin(31)) ?

The general solution for (16)/(sin(a))=(12)/(sin(31)) is a=0.75696…+360n,a=180-0.75696…+360n

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(16)/(sin(a))=(12)/(sin(31))

\frac{16}{sin ( a )} = \frac{12}{sin ( 3 1 ^{\circ} )}

a = 0.75696 \dots + 36 0^{\circ} n, a = 18 0^{\circ} - 0.75696 \dots + 36 0^{\circ} n

Radians

a = 0.75696 \dots + 2 πn, a = π - 0.75696 \dots + 2 πn

Solution steps

\frac{16}{sin ( a )} = \frac{12}{sin ( 3 1 ^{\circ} )}

Cross multiply

\frac{16}{sin ( a )} = \frac{12}{sin ( 3 1 ^{\circ} )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

16 sin (3 1^{\circ}) = sin (a) \cdot 12

16 sin (3 1^{\circ}) = sin (a) \cdot 12

Switch sides

sin (a) \cdot 12 = 16 sin (3 1^{\circ})

Divide both sides by

12

sin (a) \cdot 12 = 16 sin (3 1^{\circ})

Divide both sides by

12

\frac{sin ( a ) \cdot 12}{12} = \frac{16 sin ( 3 1 ^{\circ} )}{12}

Simplify

sin (a) = \frac{4 sin ( 3 1 ^{\circ} )}{3}

sin (a) = \frac{4 sin ( 3 1 ^{\circ} )}{3}

Verify Solutions

Find undefined (singularity) points:

sin (a) = 0

Take the denominator(s) of

\frac{16}{sin ( a )}

and compare to zero

sin (a) = 0

The following points are undefined

sin (a) = 0

Combine undefined points with solutions:

sin (a) = \frac{4 sin ( 3 1 ^{\circ} )}{3}

Apply trig inverse properties

sin (a) = \frac{4 sin ( 3 1 ^{\circ} )}{3}

General solutions for

sin (a) = \frac{4 sin ( 3 1 ^{\circ} )}{3}

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

a = arcsin (\frac{4 sin ( 3 1 ^{\circ} )}{3}) + 36 0^{\circ} n, a = 18 0^{\circ} - arcsin (\frac{4 sin ( 3 1 ^{\circ} )}{3}) + 36 0^{\circ} n

a = arcsin (\frac{4 sin ( 3 1 ^{\circ} )}{3}) + 36 0^{\circ} n, a = 18 0^{\circ} - arcsin (\frac{4 sin ( 3 1 ^{\circ} )}{3}) + 36 0^{\circ} n

Show solutions in decimal form

a = 0.75696 \dots + 36 0^{\circ} n, a = 18 0^{\circ} - 0.75696 \dots + 36 0^{\circ} n

tan (x) = \frac{25}{10}

sin (x) - 1 = 0, 0 \leq x < 2 π

3 cos (θ) = 2 cos (θ)

\frac{26.4575}{sin ( 6 0 ^{\circ} )} = \frac{20}{sin ( B )}

sin (\frac{5}{2} θ) = - 0.966

What is the general solution for (16)/(sin(a))=(12)/(sin(31)) ?
The general solution for (16)/(sin(a))=(12)/(sin(31)) is a=0.75696…+360n,a=180-0.75696…+360n