What is the general solution for 3sin(5x)=-1,x<2pi,0 ?

The general solution for 3sin(5x)=-1,x<2pi,0 is

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3sin(5x)=-1,x<2pi,0

3 sin (5 x) = - 1, x < 2 π, 0

x = - \frac{0.33983 \dots}{5}, x = \frac{π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots + 2 π}{5}, x = \frac{3 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots + 4 π}{5}, x = \frac{5 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots + 6 π}{5}, x = \frac{7 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots + 8 π}{5}, x = \frac{9 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots - 2 π}{5}, x = \frac{- π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots - 4 π}{5}, x = \frac{- 3 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots - 6 π}{5}, x = \frac{- 5 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots - 8 π}{5}, x = \frac{- 7 π + 0.33983 \dots}{5}

Degrees

x = - 3.89424 \dots^{\circ}, x = 39.89424 \dots^{\circ}, x = 68.10575 \dots^{\circ}, x = 111.89424 \dots^{\circ}, x = 140.10575 \dots^{\circ}, x = 183.89424 \dots^{\circ}, x = 212.10575 \dots^{\circ}, x = 255.89424 \dots^{\circ}, x = 284.10575 \dots^{\circ}, x = 327.89424 \dots^{\circ}, x = - 75.89424 \dots^{\circ}, x = - 32.10575 \dots^{\circ}, x = - 147.89424 \dots^{\circ}, x = - 104.10575 \dots^{\circ}, x = - 219.89424 \dots^{\circ}, x = - 176.10575 \dots^{\circ}, x = - 291.89424 \dots^{\circ}, x = - 248.10575 \dots^{\circ}

Solution steps

3 sin (5 x) = - 1, x < 2 π, 0

Divide both sides by

3

3 sin (5 x) = - 1

Divide both sides by

3

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

5 x = arcsin (- \frac{1}{3}) + 2 πn, 5 x = π + arcsin (\frac{1}{3}) + 2 πn

5 x = arcsin (- \frac{1}{3}) + 2 πn, 5 x = π + arcsin (\frac{1}{3}) + 2 πn

Solve

5 x = arcsin (- \frac{1}{3}) + 2 πn : x = - \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}

5 x = arcsin (- \frac{1}{3}) + 2 πn

Simplify

arcsin (- \frac{1}{3}) + 2 πn : - arcsin (\frac{1}{3}) + 2 πn

arcsin (- \frac{1}{3}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= - arcsin (\frac{1}{3}) + 2 πn

5 x = - arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

5

5 x = - arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

5

\frac{5 x}{5} = - \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}

Simplify

x = - \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}

x = - \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}

Solve

5 x = π + arcsin (\frac{1}{3}) + 2 πn : x = \frac{π}{5} + \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}

5 x = π + arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

5

5 x = π + arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{π}{5} + \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}

Simplify

x = \frac{π}{5} + \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}

x = \frac{π}{5} + \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}

x = - \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}, x = \frac{π}{5} + \frac{arcsin ( \frac{1}{3} )}{5} + \frac{2 πn}{5}

Solutions for the range

x < 2 π

x = - \frac{arcsin ( \frac{1}{3} )}{5}, x = \frac{π + arcsin ( \frac{1}{3} )}{5}, x = \frac{- arcsin ( \frac{1}{3} ) + 2 π}{5}, x = \frac{3 π + arcsin ( \frac{1}{3} )}{5}, x = \frac{- arcsin ( \frac{1}{3} ) + 4 π}{5}, x = \frac{5 π + arcsin ( \frac{1}{3} )}{5}, x = \frac{- arcsin ( \frac{1}{3} ) + 6 π}{5}, x = \frac{7 π + arcsin ( \frac{1}{3} )}{5}, x = \frac{- arcsin ( \frac{1}{3} ) + 8 π}{5}, x = \frac{9 π + arcsin ( \frac{1}{3} )}{5}, x = \frac{- arcsin ( \frac{1}{3} ) - 2 π}{5}, x = \frac{- π + arcsin ( \frac{1}{3} )}{5}, x = \frac{- arcsin ( \frac{1}{3} ) - 4 π}{5}, x = \frac{- 3 π + arcsin ( \frac{1}{3} )}{5}, x = \frac{- arcsin ( \frac{1}{3} ) - 6 π}{5}, x = \frac{- 5 π + arcsin ( \frac{1}{3} )}{5}, x = \frac{- arcsin ( \frac{1}{3} ) - 8 π}{5}, x = \frac{- 7 π + arcsin ( \frac{1}{3} )}{5}

Show solutions in decimal form

x = - \frac{0.33983 \dots}{5}, x = \frac{π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots + 2 π}{5}, x = \frac{3 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots + 4 π}{5}, x = \frac{5 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots + 6 π}{5}, x = \frac{7 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots + 8 π}{5}, x = \frac{9 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots - 2 π}{5}, x = \frac{- π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots - 4 π}{5}, x = \frac{- 3 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots - 6 π}{5}, x = \frac{- 5 π + 0.33983 \dots}{5}, x = \frac{- 0.33983 \dots - 8 π}{5}, x = \frac{- 7 π + 0.33983 \dots}{5}

3 cos (x) + cos (3 x) = 0

sin^{2} (a) = 2 sin (a) cos (a)

tan (x) = \frac{35}{29}

3 sin (θ) = 1 - sin (θ)

1 = sin (9 0^{\circ} - θ) - 0.075 cos (9 0^{\circ} - θ)

What is the general solution for 3sin(5x)=-1,x<2pi,0 ?
The general solution for 3sin(5x)=-1,x<2pi,0 is