What is the general solution for 589*sin(45)=443*sin(x) ?

The general solution for 589*sin(45)=443*sin(x) is x=1.22306…+360n,x=180-1.22306…+360n

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589sin(45)=443sin(x)

Solution

589 \cdot sin (4 5^{\circ}) = 443 \cdot sin (x)

Solution

x = 1.22306 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 1.22306 \dots + 36 0^{\circ} n

Radians

x = 1.22306 \dots + 2 πn, x = π - 1.22306 \dots + 2 πn

Solution steps

589 sin (4 5^{\circ}) = 443 sin (x)

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}

589 \cdot \frac{2}{2} = 443 sin (x)

Switch sides

443 sin (x) = 589 \cdot \frac{2}{2}

Divide both sides by

443

443 sin (x) = 589 \cdot \frac{2}{2}

Divide both sides by

443

\frac{443 sin ( x )}{443} = \frac{589 \cdot \frac{2}{2}}{443}

Simplify

\frac{443 sin ( x )}{443} = \frac{589 \cdot \frac{2}{2}}{443}

Simplify

\frac{443 sin ( x )}{443} : sin (x)

\frac{443 sin ( x )}{443}

Divide the numbers:

\frac{443}{443} = 1

= sin (x)

Simplify

\frac{589 \cdot \frac{2}{2}}{443} : \frac{589 2}{886}

\frac{589 \cdot \frac{2}{2}}{443}

Multiply

589 \cdot \frac{2}{2} : \frac{589}{2}

589 \cdot \frac{2}{2}

Multiply fractions:

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

= \frac{2 \cdot 589}{2}

Apply radical rule:

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

= \frac{589 \cdot 2 ^{\frac{1}{2}}}{2}

Apply exponent rule:

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

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

= \frac{589}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{589}{2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{589}{2}

= \frac{\frac{589}{2}}{443}

Apply the fraction rule:

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

= \frac{589}{2 \cdot 443}

Rationalize

\frac{589}{443 2} : \frac{589 2}{886}

\frac{589}{443 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{589 2}{2 \cdot 443 2}

2 \cdot 4432 = 886

2 \cdot 4432

Apply radical rule:

a a = a

22 = 2

= 443 \cdot 2

Multiply the numbers:

443 \cdot 2 = 886

= 886

= \frac{589 2}{886}

= \frac{589 2}{886}

sin (x) = \frac{589 2}{886}

sin (x) = \frac{589 2}{886}

sin (x) = \frac{589 2}{886}

Apply trig inverse properties

sin (x) = \frac{589 2}{886}

General solutions for

sin (x) = \frac{589 2}{886}

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

x = arcsin (\frac{589 2}{886}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{589 2}{886}) + 36 0^{\circ} n

x = arcsin (\frac{589 2}{886}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{589 2}{886}) + 36 0^{\circ} n

Show solutions in decimal form

x = 1.22306 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 1.22306 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for 589*sin(45)=443*sin(x) ?
The general solution for 589*sin(45)=443*sin(x) is x=1.22306…+360n,x=180-1.22306…+360n