What is the general solution for 15sin(30)=10sin(x) ?

The general solution for 15sin(30)=10sin(x) is x=0.84806…+360n,x=180-0.84806…+360n

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15sin(30)=10sin(x)

15 sin (3 0^{\circ}) = 10 sin (x)

x = 0.84806 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.84806 \dots + 36 0^{\circ} n

Radians

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

Solution steps

15 sin (3 0^{\circ}) = 10 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}

15 \cdot \frac{1}{2} = 10 sin (x)

Switch sides

10 sin (x) = 15 \cdot \frac{1}{2}

Divide both sides by

10

10 sin (x) = 15 \cdot \frac{1}{2}

Divide both sides by

10

\frac{10 sin ( x )}{10} = \frac{15 \cdot \frac{1}{2}}{10}

Simplify

\frac{10 sin ( x )}{10} = \frac{15 \cdot \frac{1}{2}}{10}

Simplify

\frac{10 sin ( x )}{10} : sin (x)

\frac{10 sin ( x )}{10}

Divide the numbers:

\frac{10}{10} = 1

= sin (x)

Simplify

\frac{15 \cdot \frac{1}{2}}{10} : \frac{3}{4}

\frac{15 \cdot \frac{1}{2}}{10}

Multiply

15 \cdot \frac{1}{2} : \frac{15}{2}

15 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 15}{2}

Multiply the numbers:

1 \cdot 15 = 15

= \frac{15}{2}

= \frac{\frac{15}{2}}{10}

Apply the fraction rule:

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

= \frac{15}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{15}{20}

Cancel the common factor:

5

= \frac{3}{4}

sin (x) = \frac{3}{4}

sin (x) = \frac{3}{4}

sin (x) = \frac{3}{4}

Apply trig inverse properties

sin (x) = \frac{3}{4}

General solutions for

sin (x) = \frac{3}{4}

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

x = arcsin (\frac{3}{4}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{4}) + 36 0^{\circ} n

x = arcsin (\frac{3}{4}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{4}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.84806 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.84806 \dots + 36 0^{\circ} n

What is the general solution for 15sin(30)=10sin(x) ?
The general solution for 15sin(30)=10sin(x) is x=0.84806…+360n,x=180-0.84806…+360n