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Popular Trigonometry >

(368.02)/(sin(66-x))=(290)/(sin(38))

Solution

\frac{368.02}{sin ( 6 6 ^{\circ} - x )} = \frac{290}{sin ( 3 8 ^{\circ} )}

Solution

x = - 36 0^{\circ} n + 6 6^{\circ} - 0.89673 \dots, x = - 18 0^{\circ} - 36 0^{\circ} n + 6 6^{\circ} + 0.89673 \dots

Radians

x = \frac{11 π}{30} - 0.89673 \dots - 2 πn, x = - π + \frac{11 π}{30} + 0.89673 \dots - 2 πn

Solution steps

\frac{368.02}{sin ( 6 6 ^{\circ} - x )} = \frac{290}{sin ( 3 8 ^{\circ} )}

Cross multiply

\frac{368.02}{sin ( 6 6 ^{\circ} - x )} = \frac{290}{sin ( 3 8 ^{\circ} )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

368.02 sin (3 8^{\circ}) = sin (6 6^{\circ} - x) \cdot 290

368.02 sin (3 8^{\circ}) = sin (6 6^{\circ} - x) \cdot 290

Switch sides

sin (6 6^{\circ} - x) \cdot 290 = 368.02 sin (3 8^{\circ})

Divide both sides by

290

sin (6 6^{\circ} - x) \cdot 290 = 368.02 sin (3 8^{\circ})

Divide both sides by

290

\frac{sin ( 6 6 ^{\circ} - x ) \cdot 290}{290} = \frac{368.02 sin ( 3 8 ^{\circ} )}{290}

Simplify

sin (6 6^{\circ} - x) = \frac{368.02 sin ( 3 8 ^{\circ} )}{290}

sin (6 6^{\circ} - x) = \frac{368.02 sin ( 3 8 ^{\circ} )}{290}

Verify Solutions

Find undefined (singularity) points:

sin (6 6^{\circ} - x) = 0

Take the denominator(s) of

\frac{368.02}{sin ( 6 6 ^{\circ} - x )}

and compare to zero

sin (6 6^{\circ} - x) = 0

The following points are undefined

sin (6 6^{\circ} - x) = 0

Combine undefined points with solutions:

sin (6 6^{\circ} - x) = \frac{368.02 sin ( 3 8 ^{\circ} )}{290}

Apply trig inverse properties

sin (6 6^{\circ} - x) = \frac{368.02 sin ( 3 8 ^{\circ} )}{290}

General solutions for

sin (6 6^{\circ} - x) = \frac{368.02 sin ( 3 8 ^{\circ} )}{290}

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

6 6^{\circ} - x = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n, 6 6^{\circ} - x = 18 0^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n

6 6^{\circ} - x = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n, 6 6^{\circ} - x = 18 0^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n

Solve

6 6^{\circ} - x = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n : x = - 36 0^{\circ} n + 6 6^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

6 6^{\circ} - x = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n

Move

6 6^{\circ}

to the right side

6 6^{\circ} - x = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n

Subtract

6 6^{\circ}

from both sides

6 6^{\circ} - x - 6 6^{\circ} = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n - 6 6^{\circ}

Simplify

- x = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n - 6 6^{\circ}

- x = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n - 6 6^{\circ}

Divide both sides by

- 1

- x = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n - 6 6^{\circ}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1} : - 36 0^{\circ} n + 6 6^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

\frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1}

Group like terms

= \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1} + \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1}

\frac{36 0 ^{\circ} n}{- 1} = - 36 0^{\circ} n

\frac{36 0 ^{\circ} n}{- 1}

Apply the fraction rule:

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

= - \frac{36 0 ^{\circ} n}{1}

Apply rule

\frac{a}{1} = a

= - 36 0^{\circ} n

= - 36 0^{\circ} n - \frac{6 6 ^{\circ}}{- 1} + \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1}

\frac{6 6 ^{\circ}}{- 1} = - 6 6^{\circ}

\frac{6 6 ^{\circ}}{- 1}

Apply the fraction rule:

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

= - \frac{6 6 ^{\circ}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{6 6 ^{\circ}}{1} = 6 6^{\circ}

= - 6 6^{\circ}

\frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} = - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

\frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1}

Apply the fraction rule:

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

= - \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{1} = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

= - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

= - 36 0^{\circ} n - (- 6 6^{\circ}) - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

Apply rule

- (- a) = a

= - 36 0^{\circ} n + 6 6^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

x = - 36 0^{\circ} n + 6 6^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

x = - 36 0^{\circ} n + 6 6^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

x = - 36 0^{\circ} n + 6 6^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

Solve

6 6^{\circ} - x = 18 0^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n : x = - 18 0^{\circ} - 36 0^{\circ} n + 6 6^{\circ} + arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

6 6^{\circ} - x = 18 0^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n

Move

6 6^{\circ}

to the right side

6 6^{\circ} - x = 18 0^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n

Subtract

6 6^{\circ}

from both sides

6 6^{\circ} - x - 6 6^{\circ} = 18 0^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n - 6 6^{\circ}

Simplify

- x = 18 0^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n - 6 6^{\circ}

- x = 18 0^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n - 6 6^{\circ}

Divide both sides by

- 1

- x = 18 0^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}) + 36 0^{\circ} n - 6 6^{\circ}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} - \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} - \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{18 0 ^{\circ}}{- 1} - \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1} : - 18 0^{\circ} - 36 0^{\circ} n + 6 6^{\circ} + arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

\frac{18 0 ^{\circ}}{- 1} - \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1}

Group like terms

= \frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1} - \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1}

\frac{18 0 ^{\circ}}{- 1} = - 18 0^{\circ}

\frac{18 0 ^{\circ}}{- 1}

Apply the fraction rule:

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

= - 18 0^{\circ}

Apply rule

\frac{a}{1} = a

= - 18 0^{\circ}

= - 18 0^{\circ} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 6 ^{\circ}}{- 1} - \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1}

\frac{36 0 ^{\circ} n}{- 1} = - 36 0^{\circ} n

\frac{36 0 ^{\circ} n}{- 1}

Apply the fraction rule:

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

= - \frac{36 0 ^{\circ} n}{1}

Apply rule

\frac{a}{1} = a

= - 36 0^{\circ} n

= - 18 0^{\circ} - 36 0^{\circ} n - \frac{6 6 ^{\circ}}{- 1} - \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1}

\frac{6 6 ^{\circ}}{- 1} = - 6 6^{\circ}

\frac{6 6 ^{\circ}}{- 1}

Apply the fraction rule:

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

= - \frac{6 6 ^{\circ}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{6 6 ^{\circ}}{1} = 6 6^{\circ}

= - 6 6^{\circ}

\frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1} = - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

\frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{- 1}

Apply the fraction rule:

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

= - \frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{arcsin ( \frac{368.02 s i n ( 3 8 ^{\circ} )}{290} )}{1} = arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

= - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

= - 18 0^{\circ} - 36 0^{\circ} n - (- 6 6^{\circ}) - (- arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}))

Apply rule

- (- a) = a

= - 18 0^{\circ} - 36 0^{\circ} n + 6 6^{\circ} + arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

x = - 18 0^{\circ} - 36 0^{\circ} n + 6 6^{\circ} + arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

x = - 18 0^{\circ} - 36 0^{\circ} n + 6 6^{\circ} + arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

x = - 18 0^{\circ} - 36 0^{\circ} n + 6 6^{\circ} + arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

x = - 36 0^{\circ} n + 6 6^{\circ} - arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290}), x = - 18 0^{\circ} - 36 0^{\circ} n + 6 6^{\circ} + arcsin (\frac{368.02 sin ( 3 8 ^{\circ} )}{290})

Show solutions in decimal form

x = - 36 0^{\circ} n + 6 6^{\circ} - 0.89673 \dots, x = - 18 0^{\circ} - 36 0^{\circ} n + 6 6^{\circ} + 0.89673 \dots

Graph

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