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)

Solution

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

Solution

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

Radians

x = \frac{4 \cdot 0.71851 \dots}{3} + \frac{4 π}{3} n

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

Rewrite using trig identities

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

Use the Angle Difference identity:

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

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

Simplify

sin (6 0^{\circ}) cos (\frac{3 x}{4}) - cos (6 0^{\circ}) sin (\frac{3 x}{4}) : \frac{3}{2} cos (\frac{3 x}{4}) - \frac{1}{2} sin (\frac{3 x}{4})

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

Simplify

sin (6 0^{\circ}) : \frac{3}{2}

sin (6 0^{\circ})

Use the following trivial identity:

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

Simplify

cos (6 0^{\circ}) : \frac{1}{2}

cos (6 0^{\circ})

Use the following trivial identity:

cos (6 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 (\frac{3 x}{4}) - \frac{1}{2} sin (\frac{3 x}{4})

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

Use the Angle Difference identity:

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

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

Simplify

sin (\frac{3 x}{4}) cos (3 0^{\circ}) - cos (\frac{3 x}{4}) sin (3 0^{\circ}) : \frac{3}{2} sin (\frac{3 x}{4}) - \frac{1}{2} cos (\frac{3 x}{4})

sin (\frac{3 x}{4}) cos (3 0^{\circ}) - cos (\frac{3 x}{4}) sin (3 0^{\circ})

Simplify

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

cos (3 0^{\circ})

Use the following trivial identity:

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

= \frac{3}{2} sin (\frac{3 x}{4}) - sin (3 0^{\circ}) cos (\frac{3 x}{4})

Simplify

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{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{1}{2}

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

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

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}))

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Apply rule

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

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

Divide both sides by

cos (\frac{3 x}{4}), cos (\frac{3 x}{4}) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

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

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

Move

5

to the right side

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

Subtract

5

from both sides

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

Simplify

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

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

Move

33

to the right side

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

Subtract

33

from both sides

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

Simplify

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

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

Simplify

- (3 + 53) : - 3 - 53

- (3 + 53)

Distribute parentheses

= - (3) - (53)

Apply minus-plus rules

+ (- a) = - a

= - 3 - 53

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

Divide both sides by

- 3 - 53

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

Divide both sides by

- 3 - 53

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

Simplify

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

Simplify

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

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

Cancel the common factor:

- 3 - 53

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

Simplify

- \frac{5}{- 3 - 5 3} - \frac{3 3}{- 3 - 5 3} : \frac{15 + 8 3}{33}

- \frac{5}{- 3 - 5 3} - \frac{3 3}{- 3 - 5 3}

Apply rule

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

= \frac{- 5 - 3 3}{- 3 - 5 3}

Apply the fraction rule:

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

\frac{- 5 - 3 3}{- 3 - 5 3} = \frac{- ( 5 + 3 3 )}{- ( 3 + 5 3 )}

= \frac{5 + 3 3}{3 + 5 3}

Rationalize

\frac{5 + 3 3}{3 + 5 3} : \frac{15 + 8 3}{33}

\frac{5 + 3 3}{3 + 5 3}

Multiply by the conjugate

\frac{3 - 5 3}{3 - 5 3}

= \frac{( 5 + 3 3 ) ( 3 - 5 3 )}{( 3 + 5 3 ) ( 3 - 5 3 )}

(5 + 33) (3 - 53) = - 30 - 163

(5 + 33) (3 - 53)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 5, b = 33, c = 3, d = - 53

= 5 \cdot 3 + 5 (- 53) + 33 \cdot 3 + 33 (- 53)

Apply minus-plus rules

+ (- a) = - a

= 5 \cdot 3 - 5 \cdot 53 + 3 \cdot 33 - 3 \cdot 533

Simplify

5 \cdot 3 - 5 \cdot 53 + 3 \cdot 33 - 3 \cdot 533 : - 30 - 163

5 \cdot 3 - 5 \cdot 53 + 3 \cdot 33 - 3 \cdot 533

5 \cdot 3 = 15

5 \cdot 3

Multiply the numbers:

5 \cdot 3 = 15

= 15

5 \cdot 53 = 253

5 \cdot 53

Multiply the numbers:

5 \cdot 5 = 25

= 253

3 \cdot 33 = 93

3 \cdot 33

Multiply the numbers:

3 \cdot 3 = 9

= 93

3 \cdot 533 = 45

3 \cdot 533

Multiply the numbers:

3 \cdot 5 = 15

= 1533

Apply radical rule:

a a = a

33 = 3

= 15 \cdot 3

Multiply the numbers:

15 \cdot 3 = 45

= 45

= 15 - 253 + 93 - 45

Add similar elements:

- 253 + 93 = - 163

= 15 - 163 - 45

Subtract the numbers:

15 - 45 = - 30

= - 30 - 163

= - 30 - 163

(3 + 53) (3 - 53) = - 66

(3 + 53) (3 - 53)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 3, b = 53

= 3^{2} - (53)^{2}

Simplify

3^{2} - (53)^{2} : - 66

3^{2} - (53)^{2}

3^{2} = 9

3^{2}

3^{2} = 9

= 9

(53)^{2} = 75

(53)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 5^{2} (3)^{2}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

= 5^{2} \cdot 3

5^{2} = 25

= 25 \cdot 3

Multiply the numbers:

25 \cdot 3 = 75

= 75

= 9 - 75

Subtract the numbers:

9 - 75 = - 66

= - 66

= - 66

= \frac{- 30 - 16 3}{- 66}

Apply the fraction rule:

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

- 30 - 163 = - (30 + 163)

= \frac{30 + 16 3}{66}

Factor

30 + 163 : 2 (15 + 83)

30 + 163

Rewrite as

= 2 \cdot 15 + 2 \cdot 83

Factor out common term

2

= 2 (15 + 83)

= \frac{2 ( 15 + 8 3 )}{66}

Cancel the common factor:

2

= \frac{15 + 8 3}{33}

= \frac{15 + 8 3}{33}

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

\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}

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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}

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}

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Frequently Asked Questions (FAQ)

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