What is the general solution for 1/2 sin(t)-1/2 =-1 ?

The general solution for 1/2 sin(t)-1/2 =-1 is t=(3pi)/2+2pin

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1/2 sin(t)-1/2 =-1

Solution

\frac{1}{2} sin (t) - \frac{1}{2} = - 1

Solution

t = \frac{3 π}{2} + 2 πn

Degrees

t = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{2} sin (t) - \frac{1}{2} = - 1

Move

\frac{1}{2}

to the right side

\frac{1}{2} sin (t) - \frac{1}{2} = - 1

Add

\frac{1}{2}

to both sides

\frac{1}{2} sin (t) - \frac{1}{2} + \frac{1}{2} = - 1 + \frac{1}{2}

Simplify

\frac{1}{2} sin (t) - \frac{1}{2} + \frac{1}{2} = - 1 + \frac{1}{2}

Simplify

\frac{1}{2} sin (t) - \frac{1}{2} + \frac{1}{2} : \frac{1}{2} sin (t)

\frac{1}{2} sin (t) - \frac{1}{2} + \frac{1}{2}

Add similar elements:

- \frac{1}{2} + \frac{1}{2} = 0

= \frac{1}{2} sin (t)

Simplify

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

- 1 + \frac{1}{2}

Convert element to fraction:

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

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

- 1 \cdot 2 + 1 = - 1

- 1 \cdot 2 + 1

Multiply the numbers:

1 \cdot 2 = 2

= - 2 + 1

Add/Subtract the numbers:

- 2 + 1 = - 1

= - 1

= \frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

\frac{1}{2} sin (t) = - \frac{1}{2}

\frac{1}{2} sin (t) = - \frac{1}{2}

\frac{1}{2} sin (t) = - \frac{1}{2}

Multiply both sides by

2

\frac{1}{2} sin (t) = - \frac{1}{2}

Multiply both sides by

2

2 \cdot \frac{1}{2} sin (t) = 2 (- \frac{1}{2})

Simplify

2 \cdot \frac{1}{2} sin (t) = 2 (- \frac{1}{2})

Simplify

2 \cdot \frac{1}{2} sin (t) : sin (t)

2 \cdot \frac{1}{2} sin (t)

Multiply fractions:

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

= \frac{1 \cdot 2}{2} sin (t)

Cancel the common factor:

2

= sin (t) \cdot 1

Multiply:

sin (t) \cdot 1 = sin (t)

= sin (t)

Simplify

2 (- \frac{1}{2}) : - 1

2 (- \frac{1}{2})

Remove parentheses:

(- a) = - a

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

Multiply fractions:

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

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

Cancel the common factor:

2

= - 1

sin (t) = - 1

sin (t) = - 1

sin (t) = - 1

General solutions for

sin (t) = - 1

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}

t = \frac{3 π}{2} + 2 πn

t = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

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