What is the general solution for sqrt(5)sin(θ+0.4636)=1 ?

The general solution for sqrt(5)sin(θ+0.4636)=1 is θ=0.46364…+2pin-0.4636,θ=pi-0.46364…+2pin-0.4636

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sqrt(5)sin(θ+0.4636)=1

Solution

5 sin (θ + 0.4636) = 1

Solution

θ = 0.46364 \dots + 2 πn - 0.4636, θ = π - 0.46364 \dots + 2 πn - 0.4636

Degrees

θ = 0.00272 \dots^{\circ} + 36 0^{\circ} n, θ = 126.87262 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 sin (θ + 0.4636) = 1

Divide both sides by

5

5 sin (θ + 0.4636) = 1

Divide both sides by

5

\frac{5 sin ( θ + 0.4636 )}{5} = \frac{1}{5}

Simplify

\frac{5 sin ( θ + 0.4636 )}{5} = \frac{1}{5}

Simplify

\frac{5 sin ( θ + 0.4636 )}{5} : sin (θ + 0.4636)

\frac{5 sin ( θ + 0.4636 )}{5}

Cancel the common factor:

5

= sin (θ + 0.4636)

Simplify

\frac{1}{5} : \frac{5}{5}

\frac{1}{5}

Multiply by the conjugate

\frac{5}{5}

= \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= \frac{5}{5}

sin (θ + 0.4636) = \frac{5}{5}

sin (θ + 0.4636) = \frac{5}{5}

sin (θ + 0.4636) = \frac{5}{5}

Apply trig inverse properties

sin (θ + 0.4636) = \frac{5}{5}

General solutions for

sin (θ + 0.4636) = \frac{5}{5}

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

θ + 0.4636 = arcsin (\frac{5}{5}) + 2 πn, θ + 0.4636 = π - arcsin (\frac{5}{5}) + 2 πn

θ + 0.4636 = arcsin (\frac{5}{5}) + 2 πn, θ + 0.4636 = π - arcsin (\frac{5}{5}) + 2 πn

Solve

θ + 0.4636 = arcsin (\frac{5}{5}) + 2 πn : θ = arcsin (\frac{1}{5}) + 2 πn - 0.4636

θ + 0.4636 = arcsin (\frac{5}{5}) + 2 πn

Simplify

arcsin (\frac{5}{5}) + 2 πn : arcsin (\frac{1}{5}) + 2 πn

arcsin (\frac{5}{5}) + 2 πn

\frac{5}{5} = \frac{1}{5}

\frac{5}{5}

Apply radical rule:

5 = 5^{\frac{1}{2}}

= \frac{5 ^{\frac{1}{2}}}{5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{5 ^{\frac{1}{2}}}{5 ^{1}} = \frac{1}{5 ^{1 - \frac{1}{2}}}

= \frac{1}{5 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{1}{5 ^{\frac{1}{2}}}

Apply radical rule:

5^{\frac{1}{2}} = 5

= \frac{1}{5}

= arcsin (\frac{1}{5}) + 2 πn

θ + 0.4636 = arcsin (\frac{1}{5}) + 2 πn

Move

0.4636

to the right side

θ + 0.4636 = arcsin (\frac{1}{5}) + 2 πn

Subtract

0.4636

from both sides

θ + 0.4636 - 0.4636 = arcsin (\frac{1}{5}) + 2 πn - 0.4636

Simplify

θ = arcsin (\frac{1}{5}) + 2 πn - 0.4636

θ = arcsin (\frac{1}{5}) + 2 πn - 0.4636

Solve

θ + 0.4636 = π - arcsin (\frac{5}{5}) + 2 πn : θ = π - arcsin (\frac{1}{5}) + 2 πn - 0.4636

θ + 0.4636 = π - arcsin (\frac{5}{5}) + 2 πn

Simplify

π - arcsin (\frac{5}{5}) + 2 πn : π - arcsin (\frac{1}{5}) + 2 πn

π - arcsin (\frac{5}{5}) + 2 πn

\frac{5}{5} = \frac{1}{5}

\frac{5}{5}

Apply radical rule:

5 = 5^{\frac{1}{2}}

= \frac{5 ^{\frac{1}{2}}}{5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{5 ^{\frac{1}{2}}}{5 ^{1}} = \frac{1}{5 ^{1 - \frac{1}{2}}}

= \frac{1}{5 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{1}{5 ^{\frac{1}{2}}}

Apply radical rule:

5^{\frac{1}{2}} = 5

= \frac{1}{5}

= π - arcsin (\frac{1}{5}) + 2 πn

θ + 0.4636 = π - arcsin (\frac{1}{5}) + 2 πn

Move

0.4636

to the right side

θ + 0.4636 = π - arcsin (\frac{1}{5}) + 2 πn

Subtract

0.4636

from both sides

θ + 0.4636 - 0.4636 = π - arcsin (\frac{1}{5}) + 2 πn - 0.4636

Simplify

θ = π - arcsin (\frac{1}{5}) + 2 πn - 0.4636

θ = π - arcsin (\frac{1}{5}) + 2 πn - 0.4636

θ = arcsin (\frac{1}{5}) + 2 πn - 0.4636, θ = π - arcsin (\frac{1}{5}) + 2 πn - 0.4636

Show solutions in decimal form

θ = 0.46364 \dots + 2 πn - 0.4636, θ = π - 0.46364 \dots + 2 πn - 0.4636

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(5)sin(θ+0.4636)=1 ?
The general solution for sqrt(5)sin(θ+0.4636)=1 is θ=0.46364…+2pin-0.4636,θ=pi-0.46364…+2pin-0.4636