What is the general solution for sin(x)=2*589.594*10^{-9}*300*10^3 ?

The general solution for sin(x)=2*589.594*10^{-9}*300*10^3 is x=0.36158…+2pin,x=pi-0.36158…+2pin

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Popular Trigonometry >

sin(x)=2589.59410^{-9}30010^3

Solution

sin (x) = 2 \cdot 589.594 \cdot 1 0^{- 9} \cdot 300 \cdot 1 0^{3}

Solution

x = 0.36158 \dots + 2 πn, x = π - 0.36158 \dots + 2 πn

Radians

x = 0.36158 \dots + 6.28318 \dots n, x = π - 0.36158 \dots + 6.28318 \dots n

Solution steps

sin (x) = 2 \cdot 589.594 \cdot 1 0^{- 9} \cdot 300 \cdot 1 0^{3}

Simplify

2 \cdot 589.594 \cdot 1 0^{- 9} \cdot 300 \cdot 1 0^{3} : \frac{1768.782}{5000}

2 \cdot 589.594 \cdot 1 0^{- 9} \cdot 300 \cdot 1 0^{3}

Factor integer

10 = 2 \cdot 5

= 2 \cdot 589.594 (2 \cdot 5)^{- 9} \cdot 300 \cdot 1 0^{3}

Apply exponent rule:

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

(2 \cdot 5)^{- 9} = 2^{- 9} \cdot 5^{- 9}

= 2 \cdot 589.594 \cdot 2^{- 9} \cdot 5^{- 9} \cdot 300 \cdot 1 0^{3}

Factor integer

300 = 2^{2} \cdot 5^{2} \cdot 3

= 2 \cdot 589.594 \cdot 2^{- 9} \cdot 5^{- 9} \cdot 2^{2} \cdot 5^{2} \cdot 3 \cdot 1 0^{3}

Factor integer

10 = 2 \cdot 5

= 2 \cdot 589.594 \cdot 2^{- 9} \cdot 5^{- 9} \cdot 2^{2} \cdot 5^{2} \cdot 3 (2 \cdot 5)^{3}

Apply exponent rule:

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

(2 \cdot 5)^{3} = 2^{3} \cdot 5^{3}

= 2 \cdot 589.594 \cdot 2^{- 9} \cdot 5^{- 9} \cdot 2^{2} \cdot 5^{2} \cdot 3 \cdot 2^{3} \cdot 5^{3}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2 \cdot 2^{- 9} \cdot 2^{2} \cdot 2^{3} = 2^{1 - 9 + 2 + 3}

= 589.594 \cdot 5^{- 9} \cdot 5^{2} \cdot 3 \cdot 2^{1 - 9 + 2 + 3} \cdot 5^{3}

Add/Subtract the numbers:

1 - 9 + 2 + 3 = - 3

= 589.594 \cdot 5^{- 9} \cdot 5^{2} \cdot 3 \cdot 2^{- 3} \cdot 5^{3}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

5^{- 9} \cdot 5^{2} \cdot 5^{3} = 5^{- 9 + 2 + 3}

= 589.594 \cdot 3 \cdot 2^{- 3} \cdot 5^{- 9 + 2 + 3}

Add/Subtract the numbers:

- 9 + 2 + 3 = - 4

= 589.594 \cdot 3 \cdot 2^{- 3} \cdot 5^{- 4}

Apply exponent rule:

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

2^{- 3} = \frac{1}{2 ^{3}}

= 5^{- 4} \cdot 3 \cdot 589.594 \cdot \frac{1}{2 ^{3}}

Apply exponent rule:

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

5^{- 4} = \frac{1}{5 ^{4}}

= 3 \cdot 589.594 \cdot \frac{1}{2 ^{3}} \cdot \frac{1}{5 ^{4}}

Multiply fractions:

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

= 589.594 \cdot \frac{1 \cdot 1 \cdot 3}{5 ^{4} \cdot 2 ^{3}}

\frac{1 \cdot 1 \cdot 3}{2 ^{3} \cdot 5 ^{4}} = \frac{3}{5000}

\frac{1 \cdot 1 \cdot 3}{2 ^{3} \cdot 5 ^{4}}

Multiply the numbers:

1 \cdot 1 \cdot 3 = 3

= \frac{3}{5 ^{4} \cdot 2 ^{3}}

2^{3} \cdot 5^{4} = 5000

2^{3} \cdot 5^{4}

2^{3} = 8

= 5^{4} \cdot 8

5^{4} = 625

= 8 \cdot 625

Multiply the numbers:

8 \cdot 625 = 5000

= 5000

= \frac{3}{5000}

= 589.594 \cdot \frac{3}{5000}

Multiply fractions:

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

= \frac{3 \cdot 589.594}{5000}

Multiply the numbers:

3 \cdot 589.594 = 1768.782

= \frac{1768.782}{5000}

Apply trig inverse properties

sin (x) = \frac{1768.782}{5000}

General solutions for

sin (x) = \frac{1768.782}{5000}

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

x = arcsin (\frac{1768.782}{5000}) + 2 πn, x = π - arcsin (\frac{1768.782}{5000}) + 2 πn

x = arcsin (\frac{1768.782}{5000}) + 2 πn, x = π - arcsin (\frac{1768.782}{5000}) + 2 πn

Show solutions in decimal form

x = 0.36158 \dots + 2 πn, x = π - 0.36158 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)=2*589.594*10^{-9}*300*10^3 ?
The general solution for sin(x)=2*589.594*10^{-9}*300*10^3 is x=0.36158…+2pin,x=pi-0.36158…+2pin