What is the general solution for 8sin(120-x)=9sin(x) ?

The general solution for 8sin(120-x)=9sin(x) is x=0.94566…+180n

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8sin(120-x)=9sin(x)

Solution

8 sin (12 0^{\circ} - x) = 9 sin (x)

Solution

x = 0.94566 \dots + 18 0^{\circ} n

Radians

x = 0.94566 \dots + πn

Solution steps

8 sin (12 0^{\circ} - x) = 9 sin (x)

Rewrite using trig identities

8 sin (12 0^{\circ} - x) = 9 sin (x)

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)

8 (\frac{3}{2} cos (x) + \frac{1}{2} sin (x)) = 9 sin (x)

8 (\frac{3}{2} cos (x) + \frac{1}{2} sin (x)) = 9 sin (x)

Subtract

9 sin (x)

from both sides

43 cos (x) - 5 sin (x) = 0

Rewrite using trig identities

43 cos (x) - 5 sin (x) = 0

Divide both sides by

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

\frac{4 3 cos ( x ) - 5 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

43 - \frac{5 sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

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

43 - 5 tan (x) = 0

43 - 5 tan (x) = 0

Move

43

to the right side

43 - 5 tan (x) = 0

Subtract

43

from both sides

43 - 5 tan (x) - 43 = 0 - 43

Simplify

- 5 tan (x) = - 43

- 5 tan (x) = - 43

Divide both sides by

- 5

- 5 tan (x) = - 43

Divide both sides by

- 5

\frac{- 5 tan ( x )}{- 5} = \frac{- 4 3}{- 5}

Simplify

tan (x) = \frac{4 3}{5}

tan (x) = \frac{4 3}{5}

Apply trig inverse properties

tan (x) = \frac{4 3}{5}

General solutions for

tan (x) = \frac{4 3}{5}

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

x = arctan (\frac{4 3}{5}) + 18 0^{\circ} n

x = arctan (\frac{4 3}{5}) + 18 0^{\circ} n

Show solutions in decimal form

x = 0.94566 \dots + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for 8sin(120-x)=9sin(x) ?
The general solution for 8sin(120-x)=9sin(x) is x=0.94566…+180n