What is the general solution for 1+sin(pi*t)= 3/2 ?

The general solution for 1+sin(pi*t)= 3/2 is t= 1/6+2n,t= 5/6+2n

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

Solution

1 + sin (π \cdot t) = \frac{3}{2}

Solution

t = \frac{1}{6} + 2 n, t = \frac{5}{6} + 2 n

Degrees

t = 9.54929 \dots^{\circ} + 114.59155 \dots^{\circ} n, t = 47.74648 \dots^{\circ} + 114.59155 \dots^{\circ} n

Solution steps

1 + sin (π t) = \frac{3}{2}

Move

1

to the right side

1 + sin (π t) = \frac{3}{2}

Subtract

1

from both sides

1 + sin (π t) - 1 = \frac{3}{2} - 1

Simplify

1 + sin (π t) - 1 = \frac{3}{2} - 1

Simplify

1 + sin (π t) - 1 : sin (π t)

1 + sin (π t) - 1

Add similar elements:

1 - 1 = 0

= sin (π t)

Simplify

\frac{3}{2} - 1 : \frac{1}{2}

\frac{3}{2} - 1

Convert element to fraction:

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

= - \frac{1 \cdot 2}{2} + \frac{3}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 2 + 3}{2}

- 1 \cdot 2 + 3 = 1

- 1 \cdot 2 + 3

Multiply the numbers:

1 \cdot 2 = 2

= - 2 + 3

Add/Subtract the numbers:

- 2 + 3 = 1

= 1

= \frac{1}{2}

sin (π t) = \frac{1}{2}

sin (π t) = \frac{1}{2}

sin (π t) = \frac{1}{2}

General solutions for

sin (π t) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

π t = \frac{π}{6} + 2 πn, π t = \frac{5 π}{6} + 2 πn

π t = \frac{π}{6} + 2 πn, π t = \frac{5 π}{6} + 2 πn

Solve

π t = \frac{π}{6} + 2 πn : t = \frac{1}{6} + 2 n

π t = \frac{π}{6} + 2 πn

Divide both sides by

π

π t = \frac{π}{6} + 2 πn

Divide both sides by

π

\frac{π t}{π} = \frac{\frac{π}{6}}{π} + \frac{2 πn}{π}

Simplify

\frac{π t}{π} = \frac{\frac{π}{6}}{π} + \frac{2 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{\frac{π}{6}}{π} + \frac{2 πn}{π} : \frac{1}{6} + 2 n

\frac{\frac{π}{6}}{π} + \frac{2 πn}{π}

\frac{\frac{π}{6}}{π} = \frac{1}{6}

\frac{\frac{π}{6}}{π}

Apply the fraction rule:

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

= \frac{π}{6 π}

Cancel the common factor:

π

= \frac{1}{6}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{1}{6} + 2 n

t = \frac{1}{6} + 2 n

t = \frac{1}{6} + 2 n

t = \frac{1}{6} + 2 n

Solve

π t = \frac{5 π}{6} + 2 πn : t = \frac{5}{6} + 2 n

π t = \frac{5 π}{6} + 2 πn

Divide both sides by

π

π t = \frac{5 π}{6} + 2 πn

Divide both sides by

π

\frac{π t}{π} = \frac{\frac{5 π}{6}}{π} + \frac{2 πn}{π}

Simplify

\frac{π t}{π} = \frac{\frac{5 π}{6}}{π} + \frac{2 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{\frac{5 π}{6}}{π} + \frac{2 πn}{π} : \frac{5}{6} + 2 n

\frac{\frac{5 π}{6}}{π} + \frac{2 πn}{π}

\frac{\frac{5 π}{6}}{π} = \frac{5}{6}

\frac{\frac{5 π}{6}}{π}

Apply the fraction rule:

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

= \frac{5 π}{6 π}

Cancel the common factor:

π

= \frac{5}{6}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{5}{6} + 2 n

t = \frac{5}{6} + 2 n

t = \frac{5}{6} + 2 n

t = \frac{5}{6} + 2 n

t = \frac{1}{6} + 2 n, t = \frac{5}{6} + 2 n

Popular Examples

-cos(x)=sin^2(x)-1

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

csc(x)= 5/2

csc (x) = \frac{5}{2}

csc(x)= 5/4

csc (x) = \frac{5}{4}

3cos(2θ)=9cos(θ)-6

3 cos (2 θ) = 9 cos (θ) - 6

4sin(x/3)-3=0

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

Frequently Asked Questions (FAQ)

What is the general solution for 1+sin(pi*t)= 3/2 ?
The general solution for 1+sin(pi*t)= 3/2 is t= 1/6+2n,t= 5/6+2n