What is the general solution for (15)/(sin(32))=(12)/(sin(b)) ?

The general solution for (15)/(sin(32))=(12)/(sin(b)) is b=0.43778…+360n,b=180-0.43778…+360n

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(15)/(sin(32))=(12)/(sin(b))

\frac{15}{sin ( 3 2 ^{\circ} )} = \frac{12}{sin ( b )}

b = 0.43778 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.43778 \dots + 36 0^{\circ} n

Radians

b = 0.43778 \dots + 2 πn, b = π - 0.43778 \dots + 2 πn

Solution steps

\frac{15}{sin ( 3 2 ^{\circ} )} = \frac{12}{sin ( b )}

Cross multiply

\frac{15}{sin ( 3 2 ^{\circ} )} = \frac{12}{sin ( b )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

15 sin (b) = sin (3 2^{\circ}) \cdot 12

15 sin (b) = sin (3 2^{\circ}) \cdot 12

Divide both sides by

15

15 sin (b) = sin (3 2^{\circ}) \cdot 12

Divide both sides by

15

\frac{15 sin ( b )}{15} = \frac{sin ( 3 2 ^{\circ} ) \cdot 12}{15}

Simplify

sin (b) = \frac{4 sin ( 3 2 ^{\circ} )}{5}

sin (b) = \frac{4 sin ( 3 2 ^{\circ} )}{5}

Verify Solutions

Find undefined (singularity) points:

sin (b) = 0

Take the denominator(s) of

\frac{12}{sin ( b )}

and compare to zero

sin (b) = 0

The following points are undefined

sin (b) = 0

Combine undefined points with solutions:

sin (b) = \frac{4 sin ( 3 2 ^{\circ} )}{5}

Apply trig inverse properties

sin (b) = \frac{4 sin ( 3 2 ^{\circ} )}{5}

General solutions for

sin (b) = \frac{4 sin ( 3 2 ^{\circ} )}{5}

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

b = arcsin (\frac{4 sin ( 3 2 ^{\circ} )}{5}) + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin (\frac{4 sin ( 3 2 ^{\circ} )}{5}) + 36 0^{\circ} n

b = arcsin (\frac{4 sin ( 3 2 ^{\circ} )}{5}) + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin (\frac{4 sin ( 3 2 ^{\circ} )}{5}) + 36 0^{\circ} n

Show solutions in decimal form

b = 0.43778 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.43778 \dots + 36 0^{\circ} n

\frac{sin ( x )}{25} = \frac{sin ( 12 5 ^{\circ} )}{165.6}

2 cos (3 θ) - 1 = 0

cos (4 x) = cos (8 x)

cos (2 x) - cos (x) = 2

4 sin (2 x) + 23 = 0

What is the general solution for (15)/(sin(32))=(12)/(sin(b)) ?
The general solution for (15)/(sin(32))=(12)/(sin(b)) is b=0.43778…+360n,b=180-0.43778…+360n