What is the general solution for (35)/(sin(90))=(27)/(sin(y)) ?

The general solution for (35)/(sin(90))=(27)/(sin(y)) is y=0.88108…+360n,y=180-0.88108…+360n

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(35)/(sin(90))=(27)/(sin(y))

\frac{35}{sin ( 9 0 ^{\circ} )} = \frac{27}{sin ( y )}

y = 0.88108 \dots + 36 0^{\circ} n, y = 18 0^{\circ} - 0.88108 \dots + 36 0^{\circ} n

Radians

y = 0.88108 \dots + 2 πn, y = π - 0.88108 \dots + 2 πn

Solution steps

\frac{35}{sin ( 9 0 ^{\circ} )} = \frac{27}{sin ( y )}

sin (9 0^{\circ}) = 1

sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (9 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1

\frac{35}{1} = \frac{27}{sin ( y )}

Cross multiply

\frac{35}{1} = \frac{27}{sin ( y )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

35 sin (y) = 1 \cdot 27

Simplify

1 \cdot 27 : 27

1 \cdot 27

Multiply the numbers:

1 \cdot 27 = 27

= 27

35 sin (y) = 27

35 sin (y) = 27

Divide both sides by

35

35 sin (y) = 27

Divide both sides by

35

\frac{35 sin ( y )}{35} = \frac{27}{35}

Simplify

sin (y) = \frac{27}{35}

sin (y) = \frac{27}{35}

Verify Solutions

Find undefined (singularity) points:

sin (y) = 0

Take the denominator(s) of

\frac{27}{sin ( y )}

and compare to zero

sin (y) = 0

The following points are undefined

sin (y) = 0

Combine undefined points with solutions:

sin (y) = \frac{27}{35}

Apply trig inverse properties

sin (y) = \frac{27}{35}

General solutions for

sin (y) = \frac{27}{35}

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

y = arcsin (\frac{27}{35}) + 36 0^{\circ} n, y = 18 0^{\circ} - arcsin (\frac{27}{35}) + 36 0^{\circ} n

y = arcsin (\frac{27}{35}) + 36 0^{\circ} n, y = 18 0^{\circ} - arcsin (\frac{27}{35}) + 36 0^{\circ} n

Show solutions in decimal form

y = 0.88108 \dots + 36 0^{\circ} n, y = 18 0^{\circ} - 0.88108 \dots + 36 0^{\circ} n

4 cos^{2} (x) + 2 cos (x) - 22 cos (x) = 2

sin (x + 2 0^{\circ}) \cdot csc (7 0^{\circ}) = 1

\frac{π}{2} = \frac{arcsin ( x )}{c}

6 sin^{2} (x) + 16 sin (x) cos (x) - 6 cos^{2} (x) = 0

sec^{2} (t) = 0

What is the general solution for (35)/(sin(90))=(27)/(sin(y)) ?
The general solution for (35)/(sin(90))=(27)/(sin(y)) is y=0.88108…+360n,y=180-0.88108…+360n