What is the general solution for 1.5=(sin(45))/(sin(x)) ?

The general solution for 1.5=(sin(45))/(sin(x)) is x=0.49088…+360n,x=180-0.49088…+360n

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Popular >

1.5=(sin(45))/(sin(x))

Solution

1.5 = \frac{sin ( 4 5 ^{\circ} )}{sin ( x )}

Solution

x = 0.49088 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.49088 \dots + 36 0^{\circ} n

Radians

x = 0.49088 \dots + 2 πn, x = π - 0.49088 \dots + 2 πn

Solution steps

1.5 = \frac{sin ( 4 5 ^{\circ} )}{sin ( x )}

Switch sides

\frac{sin ( 4 5 ^{\circ} )}{sin ( x )} = 1.5

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

sin (4 5^{\circ})

Use the following trivial identity:

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

sin (4 5^{\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}

= \frac{2}{2}

= \frac{2}{2}

\frac{\frac{2}{2}}{sin ( x )} = 1.5

Simplify

\frac{\frac{2}{2}}{sin ( x )} : \frac{2}{2 sin ( x )}

\frac{\frac{2}{2}}{sin ( x )}

Apply the fraction rule:

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

= \frac{2}{2 sin ( x )}

\frac{2}{2 sin ( x )} = 1.5

Multiply both sides by

sin (x)

\frac{2}{2 sin ( x )} = 1.5

Multiply both sides by

sin (x)

\frac{2}{2 sin ( x )} sin (x) = 1.5 sin (x)

Simplify

\frac{2}{2} = 1.5 sin (x)

\frac{2}{2} = 1.5 sin (x)

Switch sides

1.5 sin (x) = \frac{2}{2}

Multiply both sides by

10

1.5 sin (x) = \frac{2}{2}

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

1.5 sin (x) \cdot 10 = \frac{2}{2} \cdot 10

Refine

15 sin (x) = 52

15 sin (x) = 52

Divide both sides by

15

15 sin (x) = 52

Divide both sides by

15

\frac{15 sin ( x )}{15} = \frac{5 2}{15}

Simplify

sin (x) = \frac{2}{3}

sin (x) = \frac{2}{3}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{\frac{2}{2}}{sin ( x )}

and compare to zero

sin (x) = 0

The following points are undefined

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{2}{3}

Apply trig inverse properties

sin (x) = \frac{2}{3}

General solutions for

sin (x) = \frac{2}{3}

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

x = arcsin (\frac{2}{3}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2}{3}) + 36 0^{\circ} n

x = arcsin (\frac{2}{3}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2}{3}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.49088 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.49088 \dots + 36 0^{\circ} n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 1.5=(sin(45))/(sin(x)) ?
The general solution for 1.5=(sin(45))/(sin(x)) is x=0.49088…+360n,x=180-0.49088…+360n