What is the general solution for 3sin(60-(3x)/4)=5sin((3x)/4-30) ?

The general solution for 3sin(60-(3x)/4)=5sin((3x)/4-30) is x=(4*0.71851…)/3+(720n)/3

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3sin(60-(3x)/4)=5sin((3x)/4-30)

3 sin (6 0^{\circ} - \frac{3 x}{4}) = 5 sin (\frac{3 x}{4} - 3 0^{\circ})

x = \frac{4 \cdot 0.71851 \dots}{3} + \frac{72 0 ^{\circ} n}{3}

Solution steps

3 sin (6 0^{\circ} - \frac{3 x}{4}) = 5 sin (\frac{3 x}{4} - 3 0^{\circ})

Rewrite using trig identities

3 (\frac{3}{2} cos (\frac{3 x}{4}) - \frac{1}{2} sin (\frac{3 x}{4})) = 5 (\frac{3}{2} sin (\frac{3 x}{4}) - \frac{1}{2} cos (\frac{3 x}{4}))

Subtract

5 (\frac{3}{2} sin (\frac{3 x}{4}) - \frac{1}{2} cos (\frac{3 x}{4}))

from both sides

\frac{- 3 - 5 3}{2} sin (\frac{3 x}{4}) + \frac{5 + 3 3}{2} cos (\frac{3 x}{4}) = 0

Simplify

\frac{- 3 - 5 3}{2} sin (\frac{3 x}{4}) + \frac{5 + 3 3}{2} cos (\frac{3 x}{4}) : \frac{( - 3 - 5 3 ) sin ( \frac{3 x}{4} ) + ( 5 + 3 3 ) cos ( \frac{3 x}{4} )}{2}

\frac{( - 3 - 5 3 ) sin ( \frac{3 x}{4} ) + ( 5 + 3 3 ) cos ( \frac{3 x}{4} )}{2} = 0

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

(- 3 - 53) sin (\frac{3 x}{4}) + (5 + 33) cos (\frac{3 x}{4}) = 0

Rewrite using trig identities

- (3 + 53) tan (\frac{3 x}{4}) + 5 + 33 = 0

Move

5

to the right side

- (3 + 53) tan (\frac{3 x}{4}) + 33 = - 5

Move

33

to the right side

- (3 + 53) tan (\frac{3 x}{4}) = - 5 - 33

Simplify

- (3 + 53) : - 3 - 53

(- 3 - 53) tan (\frac{3 x}{4}) = - 5 - 33

Divide both sides by

- 3 - 53

tan (\frac{3 x}{4}) = \frac{15 + 8 3}{33}

Apply trig inverse properties

\frac{3 x}{4} = arctan (\frac{15 + 8 3}{33}) + 18 0^{\circ} n

Solve

\frac{3 x}{4} = arctan (\frac{15 + 8 3}{33}) + 18 0^{\circ} n : x = \frac{4 arctan ( \frac{15 + 8 3}{33} )}{3} + \frac{72 0 ^{\circ} n}{3}

x = \frac{4 arctan ( \frac{15 + 8 3}{33} )}{3} + \frac{72 0 ^{\circ} n}{3}

Show solutions in decimal form

x = \frac{4 \cdot 0.71851 \dots}{3} + \frac{72 0 ^{\circ} n}{3}

tan (u) = \frac{3}{3}

\frac{sin ( θ )}{60} = \frac{sin ( 8 0 ^{\circ} )}{280.87}

5.63 = 11 cos (\frac{2 π}{T} 0.137)

2 cos (x) - sin (x) = 2

x = 2 arctan (z), z

What is the general solution for 3sin(60-(3x)/4)=5sin((3x)/4-30) ?
The general solution for 3sin(60-(3x)/4)=5sin((3x)/4-30) is x=(4*0.71851…)/3+(720n)/3