What is the general solution for (125)/(sin(110))=(100)/(sin(b)) ?

The general solution for (125)/(sin(110))=(100)/(sin(b)) is b=0.85071…+360n,b=180-0.85071…+360n

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(125)/(sin(110))=(100)/(sin(b))

\frac{125}{sin ( 11 0 ^{\circ} )} = \frac{100}{sin ( b )}

b = 0.85071 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.85071 \dots + 36 0^{\circ} n

Radians

b = 0.85071 \dots + 2 πn, b = π - 0.85071 \dots + 2 πn

Solution steps

\frac{125}{sin ( 11 0 ^{\circ} )} = \frac{100}{sin ( b )}

Cross multiply

\frac{125}{sin ( 11 0 ^{\circ} )} = \frac{100}{sin ( b )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

125 sin (b) = sin (11 0^{\circ}) \cdot 100

125 sin (b) = sin (11 0^{\circ}) \cdot 100

Divide both sides by

125

125 sin (b) = sin (11 0^{\circ}) \cdot 100

Divide both sides by

125

\frac{125 sin ( b )}{125} = \frac{sin ( 11 0 ^{\circ} ) \cdot 100}{125}

Simplify

sin (b) = \frac{4 sin ( 11 0 ^{\circ} )}{5}

sin (b) = \frac{4 sin ( 11 0 ^{\circ} )}{5}

Verify Solutions

Find undefined (singularity) points:

sin (b) = 0

Take the denominator(s) of

\frac{100}{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 ( 11 0 ^{\circ} )}{5}

Apply trig inverse properties

sin (b) = \frac{4 sin ( 11 0 ^{\circ} )}{5}

General solutions for

sin (b) = \frac{4 sin ( 11 0 ^{\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 ( 11 0 ^{\circ} )}{5}) + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin (\frac{4 sin ( 11 0 ^{\circ} )}{5}) + 36 0^{\circ} n

b = arcsin (\frac{4 sin ( 11 0 ^{\circ} )}{5}) + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin (\frac{4 sin ( 11 0 ^{\circ} )}{5}) + 36 0^{\circ} n

Show solutions in decimal form

b = 0.85071 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.85071 \dots + 36 0^{\circ} n

tan (\frac{2 x}{3}) - 3 = 0

2 sin^{2} (x) - cos^{2} (x) - 2 = 0

cos (3 x) \cdot 3 = 0

16 sec^{2} (θ) - 25 = 0

2 csc^{2} (x) - 4 = 0

What is the general solution for (125)/(sin(110))=(100)/(sin(b)) ?
The general solution for (125)/(sin(110))=(100)/(sin(b)) is b=0.85071…+360n,b=180-0.85071…+360n