What is the general solution for 7/(sin(70))= 5/(sin(b)) ?

The general solution for 7/(sin(70))= 5/(sin(b)) is b=0.73583…+360n,b=180-0.73583…+360n

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7/(sin(70))= 5/(sin(b))

\frac{7}{sin ( 7 0 ^{\circ} )} = \frac{5}{sin ( b )}

b = 0.73583 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.73583 \dots + 36 0^{\circ} n

Radians

b = 0.73583 \dots + 2 πn, b = π - 0.73583 \dots + 2 πn

Solution steps

\frac{7}{sin ( 7 0 ^{\circ} )} = \frac{5}{sin ( b )}

Cross multiply

\frac{7}{sin ( 7 0 ^{\circ} )} = \frac{5}{sin ( b )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

7 sin (b) = sin (7 0^{\circ}) \cdot 5

7 sin (b) = sin (7 0^{\circ}) \cdot 5

Divide both sides by

7

7 sin (b) = sin (7 0^{\circ}) \cdot 5

Divide both sides by

7

\frac{7 sin ( b )}{7} = \frac{sin ( 7 0 ^{\circ} ) \cdot 5}{7}

Simplify

sin (b) = \frac{sin ( 7 0 ^{\circ} ) \cdot 5}{7}

sin (b) = \frac{sin ( 7 0 ^{\circ} ) \cdot 5}{7}

Verify Solutions

Find undefined (singularity) points:

sin (b) = 0

Take the denominator(s) of

\frac{5}{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{sin ( 7 0 ^{\circ} ) \cdot 5}{7}

Apply trig inverse properties

sin (b) = \frac{sin ( 7 0 ^{\circ} ) \cdot 5}{7}

General solutions for

sin (b) = \frac{sin ( 7 0 ^{\circ} ) 5}{7}

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

b = arcsin (\frac{sin ( 7 0 ^{\circ} ) \cdot 5}{7}) + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin (\frac{sin ( 7 0 ^{\circ} ) \cdot 5}{7}) + 36 0^{\circ} n

b = arcsin (\frac{sin ( 7 0 ^{\circ} ) \cdot 5}{7}) + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin (\frac{sin ( 7 0 ^{\circ} ) \cdot 5}{7}) + 36 0^{\circ} n

Show solutions in decimal form

b = 0.73583 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.73583 \dots + 36 0^{\circ} n

6 cos^{2} (x) - 7 cos (x) + 2 = 0

csc (θ) = \frac{20}{4}

sin (x + 1 0^{\circ}) = cos (x + 2 0^{\circ})

c^{2} = b^{2} + a^{2} - 2 (b) (a) cos (θ)

5 cos (θ) = 2 + cos (θ)

What is the general solution for 7/(sin(70))= 5/(sin(b)) ?
The general solution for 7/(sin(70))= 5/(sin(b)) is b=0.73583…+360n,b=180-0.73583…+360n