What is the general solution for -5pisin((pi(t))/2)=7.071 ?

The general solution for -5pisin((pi(t))/2)=7.071 is t=-(2*0.46693…)/pi+4n,t=2+(2*0.46693…)/pi+4n

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-5pisin((pi(t))/2)=7.071

Solution

- 5 π sin (\frac{π ( t )}{2}) = 7.071

Solution

t = - \frac{2 \cdot 0.46693 \dots}{π} + 4 n, t = 2 + \frac{2 \cdot 0.46693 \dots}{π} + 4 n

Degrees

t = - 17.03184 \dots^{\circ} + 229.18311 \dots^{\circ} n, t = 131.62340 \dots^{\circ} + 229.18311 \dots^{\circ} n

Solution steps

- 5 π sin (\frac{π ( t )}{2}) = 7.071

Divide both sides by

- 5 π

- 5 π sin (\frac{π t}{2}) = 7.071

Divide both sides by

- 5 π

\frac{- 5 π sin ( \frac{π t}{2} )}{- 5 π} = \frac{7.071}{- 5 π}

Simplify

\frac{- 5 π sin ( \frac{π t}{2} )}{- 5 π} = \frac{7.071}{- 5 π}

Simplify

\frac{- 5 π sin ( \frac{π t}{2} )}{- 5 π} : sin (\frac{π t}{2})

\frac{- 5 π sin ( \frac{π t}{2} )}{- 5 π}

Apply the fraction rule:

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

= \frac{5 π sin ( \frac{π t}{2} )}{5 π}

Divide the numbers:

\frac{5}{5} = 1

= \frac{π sin ( \frac{π t}{2} )}{π}

Cancel the common factor:

π

= sin (\frac{π t}{2})

Simplify

\frac{7.071}{- 5 π} : - 0.45015 \dots

\frac{7.071}{- 5 π}

Apply the fraction rule:

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

= - \frac{7.071}{5 π}

Simplify

\frac{7.071}{5 π} : 0.45015 \dots

\frac{7.071}{5 π}

Multiply the numbers:

5 \cdot 3.14159 \dots = 15.70796 \dots

= \frac{7.071}{15.70796 \dots}

Divide the numbers:

\frac{7.071}{15.70796 \dots} = 0.45015 \dots

= 0.45015 \dots

= - 0.45015 \dots

sin (\frac{π t}{2}) = - 0.45015 \dots

sin (\frac{π t}{2}) = - 0.45015 \dots

sin (\frac{π t}{2}) = - 0.45015 \dots

Apply trig inverse properties

sin (\frac{π t}{2}) = - 0.45015 \dots

General solutions for

sin (\frac{π t}{2}) = - 0.45015 \dots

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

\frac{π t}{2} = arcsin (- 0.45015 \dots) + 2 πn, \frac{π t}{2} = π + arcsin (0.45015 \dots) + 2 πn

\frac{π t}{2} = arcsin (- 0.45015 \dots) + 2 πn, \frac{π t}{2} = π + arcsin (0.45015 \dots) + 2 πn

Solve

\frac{π t}{2} = arcsin (- 0.45015 \dots) + 2 πn : t = - \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

\frac{π t}{2} = arcsin (- 0.45015 \dots) + 2 πn

Simplify

arcsin (- 0.45015 \dots) + 2 πn : - arcsin (0.45015 \dots) + 2 πn

arcsin (- 0.45015 \dots) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 0.45015 \dots) = - arcsin (0.45015 \dots)

= - arcsin (0.45015 \dots) + 2 πn

\frac{π t}{2} = - arcsin (0.45015 \dots) + 2 πn

Multiply both sides by

2

\frac{π t}{2} = - arcsin (0.45015 \dots) + 2 πn

Multiply both sides by

2

\frac{2 π t}{2} = - 2 arcsin (0.45015 \dots) + 2 \cdot 2 πn

Simplify

π t = - 2 arcsin (0.45015 \dots) + 4 πn

π t = - 2 arcsin (0.45015 \dots) + 4 πn

Divide both sides by

π

π t = - 2 arcsin (0.45015 \dots) + 4 πn

Divide both sides by

π

\frac{π t}{π} = - \frac{2 arcsin ( 0.45015 \dots )}{π} + \frac{4 πn}{π}

Simplify

t = - \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

t = - \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

Solve

\frac{π t}{2} = π + arcsin (0.45015 \dots) + 2 πn : t = 2 + \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

\frac{π t}{2} = π + arcsin (0.45015 \dots) + 2 πn

Multiply both sides by

2

\frac{π t}{2} = π + arcsin (0.45015 \dots) + 2 πn

Multiply both sides by

2

\frac{2 π t}{2} = 2 π + 2 arcsin (0.45015 \dots) + 2 \cdot 2 πn

Simplify

π t = 2 π + 2 arcsin (0.45015 \dots) + 4 πn

π t = 2 π + 2 arcsin (0.45015 \dots) + 4 πn

Divide both sides by

π

π t = 2 π + 2 arcsin (0.45015 \dots) + 4 πn

Divide both sides by

π

\frac{π t}{π} = \frac{2 π}{π} + \frac{2 arcsin ( 0.45015 \dots )}{π} + \frac{4 πn}{π}

Simplify

\frac{π t}{π} = \frac{2 π}{π} + \frac{2 arcsin ( 0.45015 \dots )}{π} + \frac{4 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{2 π}{π} + \frac{2 arcsin ( 0.45015 \dots )}{π} + \frac{4 πn}{π} : 2 + \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

\frac{2 π}{π} + \frac{2 arcsin ( 0.45015 \dots )}{π} + \frac{4 πn}{π}

Cancel

\frac{2 π}{π} : 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

= 2 + \frac{2 arcsin ( 0.45015 \dots )}{π} + \frac{4 πn}{π}

Cancel

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= 2 + \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

t = 2 + \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

t = 2 + \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

t = 2 + \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

t = - \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n, t = 2 + \frac{2 arcsin ( 0.45015 \dots )}{π} + 4 n

Show solutions in decimal form

t = - \frac{2 \cdot 0.46693 \dots}{π} + 4 n, t = 2 + \frac{2 \cdot 0.46693 \dots}{π} + 4 n

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Frequently Asked Questions (FAQ)

What is the general solution for -5pisin((pi(t))/2)=7.071 ?
The general solution for -5pisin((pi(t))/2)=7.071 is t=-(2*0.46693…)/pi+4n,t=2+(2*0.46693…)/pi+4n