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

The general solution for sin(3x)=(-3)/4 is x=-(0.84806…)/3+(2pin)/3 ,x= pi/3+(0.84806…)/3+(2pin)/3

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sin(3x)=(-3)/4

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

x = - \frac{0.84806 \dots}{3} + \frac{2 πn}{3}, x = \frac{π}{3} + \frac{0.84806 \dots}{3} + \frac{2 πn}{3}

Degrees

x = - 16.19679 \dots^{\circ} + 12 0^{\circ} n, x = 76.19679 \dots^{\circ} + 12 0^{\circ} n

Solution steps

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

Apply the fraction rule:

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

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

Apply trig inverse properties

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

General solutions for

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

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

3 x = arcsin (- \frac{3}{4}) + 2 πn, 3 x = π + arcsin (\frac{3}{4}) + 2 πn

3 x = arcsin (- \frac{3}{4}) + 2 πn, 3 x = π + arcsin (\frac{3}{4}) + 2 πn

Solve

3 x = arcsin (- \frac{3}{4}) + 2 πn : x = - \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}

3 x = arcsin (- \frac{3}{4}) + 2 πn

Simplify

arcsin (- \frac{3}{4}) + 2 πn : - arcsin (\frac{3}{4}) + 2 πn

arcsin (- \frac{3}{4}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{4}) = - arcsin (\frac{3}{4})

= - arcsin (\frac{3}{4}) + 2 πn

3 x = - arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

3

3 x = - arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = - \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}

Simplify

x = - \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}

x = - \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}

Solve

3 x = π + arcsin (\frac{3}{4}) + 2 πn : x = \frac{π}{3} + \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}

3 x = π + arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

3

3 x = π + arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{π}{3} + \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}

Simplify

x = \frac{π}{3} + \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}

x = - \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}, x = \frac{π}{3} + \frac{arcsin ( \frac{3}{4} )}{3} + \frac{2 πn}{3}

Show solutions in decimal form

x = - \frac{0.84806 \dots}{3} + \frac{2 πn}{3}, x = \frac{π}{3} + \frac{0.84806 \dots}{3} + \frac{2 πn}{3}

4 cos (x - 4 5^{\circ}) + 7 = 10

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

- 2 = 2 cos (3 θ)

cos (x) = \frac{4}{30}

t, x = sin (2 t) + 1

What is the general solution for sin(3x)=(-3)/4 ?
The general solution for sin(3x)=(-3)/4 is x=-(0.84806…)/3+(2pin)/3 ,x= pi/3+(0.84806…)/3+(2pin)/3