What is the general solution for pi/3 =3+2sin(2t) ?

The general solution for pi/3 =3+2sin(2t) is t=(-1.35311…)/2+pin,t= pi/2+(1.35311…)/2+pin

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pi/3 =3+2sin(2t)

Solution

\frac{π}{3} = 3 + 2 sin (2 t)

Solution

t = \frac{- 1.35311 \dots}{2} + πn, t = \frac{π}{2} + \frac{1.35311 \dots}{2} + πn

Degrees

t = - 38.76394 \dots^{\circ} + 18 0^{\circ} n, t = 128.76394 \dots^{\circ} + 18 0^{\circ} n

Solution steps

\frac{π}{3} = 3 + 2 sin (2 t)

Switch sides

3 + 2 sin (2 t) = \frac{π}{3}

Move

3

to the right side

3 + 2 sin (2 t) = \frac{π}{3}

Subtract

3

from both sides

3 + 2 sin (2 t) - 3 = \frac{π}{3} - 3

Simplify

2 sin (2 t) = \frac{π}{3} - 3

2 sin (2 t) = \frac{π}{3} - 3

Divide both sides by

2

2 sin (2 t) = \frac{π}{3} - 3

Divide both sides by

2

\frac{2 sin ( 2 t )}{2} = \frac{\frac{π}{3}}{2} - \frac{3}{2}

Simplify

\frac{2 sin ( 2 t )}{2} = \frac{\frac{π}{3}}{2} - \frac{3}{2}

Simplify

\frac{2 sin ( 2 t )}{2} : sin (2 t)

\frac{2 sin ( 2 t )}{2}

Divide the numbers:

\frac{2}{2} = 1

= sin (2 t)

Simplify

\frac{\frac{π}{3}}{2} - \frac{3}{2} : \frac{π - 9}{6}

\frac{\frac{π}{3}}{2} - \frac{3}{2}

Apply rule

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

= \frac{\frac{π}{3} - 3}{2}

Join

\frac{π}{3} - 3 : \frac{π - 9}{3}

\frac{π}{3} - 3

Convert element to fraction:

3 = \frac{3 \cdot 3}{3}

= \frac{π}{3} - \frac{3 \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π - 3 \cdot 3}{3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π - 9}{3}

= \frac{\frac{π - 9}{3}}{2}

Apply the fraction rule:

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

= \frac{π - 9}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π - 9}{6}

sin (2 t) = \frac{π - 9}{6}

sin (2 t) = \frac{π - 9}{6}

sin (2 t) = \frac{π - 9}{6}

Apply trig inverse properties

sin (2 t) = \frac{π - 9}{6}

General solutions for

sin (2 t) = \frac{π - 9}{6}

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

2 t = arcsin (\frac{π - 9}{6}) + 2 πn, 2 t = π + arcsin (- \frac{π - 9}{6}) + 2 πn

2 t = arcsin (\frac{π - 9}{6}) + 2 πn, 2 t = π + arcsin (- \frac{π - 9}{6}) + 2 πn

Solve

2 t = arcsin (\frac{π - 9}{6}) + 2 πn : t = \frac{arcsin ( \frac{π - 9}{6} )}{2} + πn

2 t = arcsin (\frac{π - 9}{6}) + 2 πn

Divide both sides by

2

2 t = arcsin (\frac{π - 9}{6}) + 2 πn

Divide both sides by

2

\frac{2 t}{2} = \frac{arcsin ( \frac{π - 9}{6} )}{2} + \frac{2 πn}{2}

Simplify

t = \frac{arcsin ( \frac{π - 9}{6} )}{2} + πn

t = \frac{arcsin ( \frac{π - 9}{6} )}{2} + πn

Solve

2 t = π + arcsin (- \frac{π - 9}{6}) + 2 πn : t = \frac{π}{2} + \frac{arcsin ( \frac{- π + 9}{6} )}{2} + πn

2 t = π + arcsin (- \frac{π - 9}{6}) + 2 πn

- \frac{π - 9}{6} = - \frac{- ( - π + 9 )}{6} = \frac{- π + 9}{6}

2 t = π + arcsin (\frac{- π + 9}{6}) + 2 πn

Divide both sides by

2

2 t = π + arcsin (\frac{- π + 9}{6}) + 2 πn

Divide both sides by

2

\frac{2 t}{2} = \frac{π}{2} + \frac{arcsin ( \frac{- π + 9}{6} )}{2} + \frac{2 πn}{2}

Simplify

t = \frac{π}{2} + \frac{arcsin ( \frac{- π + 9}{6} )}{2} + πn

t = \frac{π}{2} + \frac{arcsin ( \frac{- π + 9}{6} )}{2} + πn

t = \frac{arcsin ( \frac{π - 9}{6} )}{2} + πn, t = \frac{π}{2} + \frac{arcsin ( \frac{- π + 9}{6} )}{2} + πn

Show solutions in decimal form

t = \frac{- 1.35311 \dots}{2} + πn, t = \frac{π}{2} + \frac{1.35311 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for pi/3 =3+2sin(2t) ?
The general solution for pi/3 =3+2sin(2t) is t=(-1.35311…)/2+pin,t= pi/2+(1.35311…)/2+pin