What is the general solution for (sin(pi/2))/5 =(sin(x))/4 ?

The general solution for (sin(pi/2))/5 =(sin(x))/4 is x=0.92729…+2pin,x=pi-0.92729…+2pin

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(sin(pi/2))/5 =(sin(x))/4

\frac{sin ( \frac{π}{2} )}{5} = \frac{sin ( x )}{4}

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn

Degrees

x = 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{sin ( \frac{π}{2} )}{5} = \frac{sin ( x )}{4}

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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{1}{5} = \frac{sin ( x )}{4}

Switch sides

\frac{sin ( x )}{4} = \frac{1}{5}

Multiply both sides by

4

\frac{sin ( x )}{4} = \frac{1}{5}

Multiply both sides by

4

\frac{4 sin ( x )}{4} = \frac{1 \cdot 4}{5}

Simplify

\frac{4 sin ( x )}{4} = \frac{1 \cdot 4}{5}

Simplify

\frac{4 sin ( x )}{4} : sin (x)

\frac{4 sin ( x )}{4}

Divide the numbers:

\frac{4}{4} = 1

= sin (x)

Simplify

\frac{1 \cdot 4}{5} : \frac{4}{5}

\frac{1 \cdot 4}{5}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{5}

sin (x) = \frac{4}{5}

sin (x) = \frac{4}{5}

sin (x) = \frac{4}{5}

Apply trig inverse properties

sin (x) = \frac{4}{5}

General solutions for

sin (x) = \frac{4}{5}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn

x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn

Show solutions in decimal form

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn

What is the general solution for (sin(pi/2))/5 =(sin(x))/4 ?
The general solution for (sin(pi/2))/5 =(sin(x))/4 is x=0.92729…+2pin,x=pi-0.92729…+2pin