What is the general solution for solvefor a,sin(2a)+1=0.5 ?

The general solution for solvefor a,sin(2a)+1=0.5 is a=(7pi)/(12)+pin,a=(11pi)/(12)+pin

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solvefor a,sin(2a)+1=0.5

Solution

solve for

a, sin (2 a) + 1 = 0.5

Solution

a = \frac{7 π}{12} + πn, a = \frac{11 π}{12} + πn

Degrees

a = 10 5^{\circ} + 18 0^{\circ} n, a = 16 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (2 a) + 1 = 0.5

Move

1

to the right side

sin (2 a) + 1 = 0.5

Subtract

1

from both sides

sin (2 a) + 1 - 1 = 0.5 - 1

Simplify

sin (2 a) = - 0.5

sin (2 a) = - 0.5

General solutions for

sin (2 a) = - 0.5

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}

2 a = \frac{7 π}{6} + 2 πn, 2 a = \frac{11 π}{6} + 2 πn

2 a = \frac{7 π}{6} + 2 πn, 2 a = \frac{11 π}{6} + 2 πn

Solve

2 a = \frac{7 π}{6} + 2 πn : a = \frac{7 π}{12} + πn

2 a = \frac{7 π}{6} + 2 πn

Divide both sides by

2

2 a = \frac{7 π}{6} + 2 πn

Divide both sides by

2

\frac{2 a}{2} = \frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 a}{2} = \frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 a}{2} : a

\frac{2 a}{2}

Divide the numbers:

\frac{2}{2} = 1

= a

Simplify

\frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2} : \frac{7 π}{12} + πn

\frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{6}}{2} = \frac{7 π}{12}

\frac{\frac{7 π}{6}}{2}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{7 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{7 π}{12} + πn

a = \frac{7 π}{12} + πn

a = \frac{7 π}{12} + πn

a = \frac{7 π}{12} + πn

Solve

2 a = \frac{11 π}{6} + 2 πn : a = \frac{11 π}{12} + πn

2 a = \frac{11 π}{6} + 2 πn

Divide both sides by

2

2 a = \frac{11 π}{6} + 2 πn

Divide both sides by

2

\frac{2 a}{2} = \frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 a}{2} = \frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 a}{2} : a

\frac{2 a}{2}

Divide the numbers:

\frac{2}{2} = 1

= a

Simplify

\frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2} : \frac{11 π}{12} + πn

\frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{11 π}{6}}{2} = \frac{11 π}{12}

\frac{\frac{11 π}{6}}{2}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{11 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{11 π}{12} + πn

a = \frac{11 π}{12} + πn

a = \frac{11 π}{12} + πn

a = \frac{11 π}{12} + πn

a = \frac{7 π}{12} + πn, a = \frac{11 π}{12} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor a,sin(2a)+1=0.5 ?
The general solution for solvefor a,sin(2a)+1=0.5 is a=(7pi)/(12)+pin,a=(11pi)/(12)+pin