What is the general solution for 0.5*sin(x)=1*(645*10^{-9}) ?

The general solution for 0.5*sin(x)=1*(645*10^{-9}) is x=0.00000129+2pin,x=pi-0.00000129+2pin

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0.5sin(x)=1(645*10^{-9})

Solution

0.5 \cdot sin (x) = 1 \cdot (645 \cdot 1 0^{- 9})

Solution

x = 0.00000129 + 2 πn, x = π - 0.00000129 + 2 πn

Degrees

x = 0.00007 \dots^{\circ} + 36 0^{\circ} n, x = 179.99992 \dots^{\circ} + 36 0^{\circ} n

Solution steps

0.5 sin (x) = 1 \cdot (645 \cdot 1 0^{- 9})

Simplify

1 \cdot 645 \cdot 1 0^{- 9} : \frac{129}{200000000}

1 \cdot 645 \cdot 1 0^{- 9}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

1 0^{- 9} = \frac{1}{1 0 ^{9}}

= 1 \cdot 645 \cdot \frac{1}{1 0 ^{9}}

Multiply fractions:

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

= 1 \cdot \frac{1 \cdot 645}{1 0 ^{9}}

\frac{1 \cdot 645}{1 0 ^{9}} = \frac{129}{200000000}

\frac{1 \cdot 645}{1 0 ^{9}}

Multiply the numbers:

1 \cdot 645 = 645

= \frac{645}{1 0 ^{9}}

Factor

645 : 3 \cdot 5 \cdot 43

Factor

645 = 3 \cdot 5 \cdot 43

Factor

1 0^{9} : 2^{9} \cdot 5^{9}

Factor

10 = 2 \cdot 5

= (2 \cdot 5)^{9}

Apply exponent rule:

(ab)^{c} = a^{c} b^{c}

= 2^{9} \cdot 5^{9}

= \frac{3 \cdot 5 \cdot 43}{2 ^{9} \cdot 5 ^{9}}

Cancel the common factor:

5

= \frac{3 \cdot 43}{2 ^{9} \cdot 5 ^{8}}

Multiply the numbers:

3 \cdot 43 = 129

= \frac{129}{2 ^{9} \cdot 5 ^{8}}

2^{9} \cdot 5^{8} = 200000000

2^{9} \cdot 5^{8}

2^{9} = 512

= 5^{8} \cdot 512

5^{8} = 390625

= 512 \cdot 390625

Multiply the numbers:

512 \cdot 390625 = 200000000

= 200000000

= \frac{129}{200000000}

= 1 \cdot \frac{129}{200000000}

Multiply:

1 \cdot \frac{129}{200000000} = \frac{129}{200000000}

= \frac{129}{200000000}

0.5 sin (x) = \frac{129}{200000000}

Multiply both sides by

10

0.5 sin (x) = \frac{129}{200000000}

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

0.5 sin (x) \cdot 10 = \frac{129}{200000000} \cdot 10

Refine

5 sin (x) = \frac{129}{20000000}

5 sin (x) = \frac{129}{20000000}

Divide both sides by

5

5 sin (x) = \frac{129}{20000000}

Divide both sides by

5

\frac{5 sin ( x )}{5} = \frac{\frac{129}{20000000}}{5}

Simplify

\frac{5 sin ( x )}{5} = \frac{\frac{129}{20000000}}{5}

Simplify

\frac{5 sin ( x )}{5} : sin (x)

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

Divide the numbers:

\frac{5}{5} = 1

= sin (x)

Simplify

\frac{\frac{129}{20000000}}{5} : \frac{129}{100000000}

\frac{\frac{129}{20000000}}{5}

Apply the fraction rule:

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

= \frac{129}{20000000 \cdot 5}

Multiply the numbers:

20000000 \cdot 5 = 100000000

= \frac{129}{100000000}

sin (x) = \frac{129}{100000000}

sin (x) = \frac{129}{100000000}

sin (x) = \frac{129}{100000000}

Apply trig inverse properties

sin (x) = \frac{129}{100000000}

General solutions for

sin (x) = \frac{129}{100000000}

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

x = arcsin (\frac{129}{100000000}) + 2 πn, x = π - arcsin (\frac{129}{100000000}) + 2 πn

x = arcsin (\frac{129}{100000000}) + 2 πn, x = π - arcsin (\frac{129}{100000000}) + 2 πn

Show solutions in decimal form

x = 0.00000129 + 2 πn, x = π - 0.00000129 + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 0.5*sin(x)=1*(645*10^{-9}) ?
The general solution for 0.5*sin(x)=1*(645*10^{-9}) is x=0.00000129+2pin,x=pi-0.00000129+2pin