What is the general solution for 1.25sin(30)=1.5sin(x) ?

The general solution for 1.25sin(30)=1.5sin(x) is x=0.42977…+360n,x=180-0.42977…+360n

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1.25sin(30)=1.5sin(x)

1.25 sin (3 0^{\circ}) = 1.5 sin (x)

x = 0.42977 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.42977 \dots + 36 0^{\circ} n

Radians

x = 0.42977 \dots + 2 πn, x = π - 0.42977 \dots + 2 πn

Solution steps

1.25 sin (3 0^{\circ}) = 1.5 sin (x)

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 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}

= \frac{1}{2}

= \frac{1}{2}

1.25 \cdot \frac{1}{2} = 1.5 sin (x)

Switch sides

1.5 sin (x) = 1.25 \cdot \frac{1}{2}

Divide both sides by

1.5

1.5 sin (x) = 1.25 \cdot \frac{1}{2}

Divide both sides by

1.5

\frac{1.5 sin ( x )}{1.5} = \frac{1.25 \cdot \frac{1}{2}}{1.5}

Simplify

\frac{1.5 sin ( x )}{1.5} = \frac{1.25 \cdot \frac{1}{2}}{1.5}

Simplify

\frac{1.5 sin ( x )}{1.5} : sin (x)

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

Cancel the common factor:

1.5

= sin (x)

Simplify

\frac{1.25 \cdot \frac{1}{2}}{1.5} : 0.41666 \dots

\frac{1.25 \cdot \frac{1}{2}}{1.5}

Multiply

1.25 \cdot \frac{1}{2} : 0.625

1.25 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 1.25}{2}

Multiply the numbers:

1 \cdot 1.25 = 1.25

= \frac{1.25}{2}

Divide the numbers:

\frac{1.25}{2} = 0.625

= 0.625

= \frac{0.625}{1.5}

Divide the numbers:

\frac{0.625}{1.5} = 0.41666 \dots

= 0.41666 \dots

sin (x) = 0.41666 \dots

sin (x) = 0.41666 \dots

sin (x) = 0.41666 \dots

Apply trig inverse properties

sin (x) = 0.41666 \dots

General solutions for

sin (x) = 0.41666 \dots

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

x = arcsin (0.41666 \dots) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (0.41666 \dots) + 36 0^{\circ} n

x = arcsin (0.41666 \dots) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (0.41666 \dots) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.42977 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.42977 \dots + 36 0^{\circ} n

What is the general solution for 1.25sin(30)=1.5sin(x) ?
The general solution for 1.25sin(30)=1.5sin(x) is x=0.42977…+360n,x=180-0.42977…+360n