What is the general solution for 50/33 =(sin(x))/(sin(120-x)) ?

The general solution for 50/33 =(sin(x))/(sin(120-x)) is x=1.38810…+180n

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50/33 =(sin(x))/(sin(120-x))

Solution

\frac{50}{33} = \frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )}

Solution

x = 1.38810 \dots + 18 0^{\circ} n

Radians

x = 1.38810 \dots + πn

Solution steps

\frac{50}{33} = \frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )}

Switch sides

\frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )} = \frac{50}{33}

Rewrite using trig identities

\frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )} = \frac{50}{33}

Rewrite using trig identities

sin (12 0^{\circ} - x)

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (12 0^{\circ}) cos (x) - cos (12 0^{\circ}) sin (x)

Simplify

sin (12 0^{\circ}) cos (x) - cos (12 0^{\circ}) sin (x) : \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

sin (12 0^{\circ}) cos (x) - cos (12 0^{\circ}) sin (x)

Simplify

sin (12 0^{\circ}) : \frac{3}{2}

sin (12 0^{\circ})

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

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{3}{2}

= \frac{3}{2} cos (x) - cos (12 0^{\circ}) sin (x)

Simplify

cos (12 0^{\circ}) : - \frac{1}{2}

cos (12 0^{\circ})

Use the following trivial identity:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{1}{2}

= \frac{3}{2} cos (x) - (- \frac{1}{2} sin (x))

Apply rule

- (- a) = a

= \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

= \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} = \frac{50}{33}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} = \frac{50}{33}

Subtract

\frac{50}{33}

from both sides

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} - \frac{50}{33} = 0

Simplify

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} - \frac{50}{33} : \frac{16 sin ( x ) - 50 3 cos ( x )}{33 ( 3 cos ( x ) + sin ( x ) )}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} - \frac{50}{33}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} = \frac{2 sin ( x )}{3 cos ( x ) + sin ( x )}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}

\frac{3}{2} cos (x) = \frac{3 cos ( x )}{2}

\frac{3}{2} cos (x)

Multiply fractions:

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

= \frac{3 cos ( x )}{2}

\frac{1}{2} sin (x) = \frac{sin ( x )}{2}

\frac{1}{2} sin (x)

Multiply fractions:

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

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

Multiply:

1 \cdot sin (x) = sin (x)

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

= \frac{sin ( x )}{\frac{3 c o s ( x )}{2} + \frac{s i n ( x )}{2}}

Combine the fractions

\frac{3 cos ( x )}{2} + \frac{sin ( x )}{2} : \frac{3 cos ( x ) + sin ( x )}{2}

Apply rule

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

= \frac{3 cos ( x ) + sin ( x )}{2}

= \frac{sin ( x )}{\frac{3 c o s ( x ) + s i n ( x )}{2}}

Apply the fraction rule:

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

= \frac{sin ( x ) \cdot 2}{3 cos ( x ) + sin ( x )}

= \frac{2 sin ( x )}{3 cos ( x ) + sin ( x )} - \frac{50}{33}

Least Common Multiplier of

3 cos (x) + sin (x), 33 : 33 (3 cos (x) + sin (x))

3 cos (x) + sin (x), 33

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

3 cos (x) + sin (x)

33

= 33 (3 cos (x) + sin (x))

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

33 (3 cos (x) + sin (x))

For

\frac{sin ( x ) \cdot 2}{3 cos ( x ) + sin ( x )} :

multiply the denominator and numerator by

33

\frac{sin ( x ) \cdot 2}{3 cos ( x ) + sin ( x )} = \frac{sin ( x ) \cdot 2 \cdot 33}{( 3 cos ( x ) + sin ( x ) ) \cdot 33} = \frac{66 sin ( x )}{33 ( 3 cos ( x ) + sin ( x ) )}

For

\frac{50}{33} :

multiply the denominator and numerator by

3 cos (x) + sin (x)

\frac{50}{33} = \frac{50 ( 3 cos ( x ) + sin ( x ) )}{33 ( 3 cos ( x ) + sin ( x ) )}

= \frac{66 sin ( x )}{33 ( 3 cos ( x ) + sin ( x ) )} - \frac{50 ( 3 cos ( x ) + sin ( x ) )}{33 ( 3 cos ( x ) + sin ( x ) )}

Since the denominators are equal, combine the fractions:

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

= \frac{66 sin ( x ) - 50 ( 3 cos ( x ) + sin ( x ) )}{33 ( 3 cos ( x ) + sin ( x ) )}

Expand

66 sin (x) - 50 (3 cos (x) + sin (x)) : 16 sin (x) - 503 cos (x)

66 sin (x) - 50 (3 cos (x) + sin (x))

Expand

- 50 (3 cos (x) + sin (x)) : - 503 cos (x) - 50 sin (x)

- 50 (3 cos (x) + sin (x))

Apply the distributive law:

a (b + c) = ab + a c

a = - 50, b = 3 cos (x), c = sin (x)

= - 503 cos (x) + (- 50) sin (x)

Apply minus-plus rules

+ (- a) = - a

= - 503 cos (x) - 50 sin (x)

= 66 sin (x) - 503 cos (x) - 50 sin (x)

Add similar elements:

66 sin (x) - 50 sin (x) = 16 sin (x)

= 16 sin (x) - 503 cos (x)

= \frac{16 sin ( x ) - 50 3 cos ( x )}{33 ( 3 cos ( x ) + sin ( x ) )}

\frac{16 sin ( x ) - 50 3 cos ( x )}{33 ( 3 cos ( x ) + sin ( x ) )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

16 sin (x) - 503 cos (x) = 0

Rewrite using trig identities

16 sin (x) - 503 cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{16 sin ( x ) - 50 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{16 sin ( x )}{cos ( x )} - 503 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

16 tan (x) - 503 = 0

16 tan (x) - 503 = 0

Move

503

to the right side

16 tan (x) - 503 = 0

Add

503

to both sides

16 tan (x) - 503 + 503 = 0 + 503

Simplify

16 tan (x) = 503

16 tan (x) = 503

Divide both sides by

16

16 tan (x) = 503

Divide both sides by

16

\frac{16 tan ( x )}{16} = \frac{50 3}{16}

Simplify

tan (x) = \frac{25 3}{8}

tan (x) = \frac{25 3}{8}

Apply trig inverse properties

tan (x) = \frac{25 3}{8}

General solutions for

tan (x) = \frac{25 3}{8}

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

x = arctan (\frac{25 3}{8}) + 18 0^{\circ} n

x = arctan (\frac{25 3}{8}) + 18 0^{\circ} n

Show solutions in decimal form

x = 1.38810 \dots + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for 50/33 =(sin(x))/(sin(120-x)) ?
The general solution for 50/33 =(sin(x))/(sin(120-x)) is x=1.38810…+180n