What is the general solution for sin(t)=(250)/(sqrt(250^2+300^2)) ?

The general solution for sin(t)=(250)/(sqrt(250^2+300^2)) is t=0.69473…+2pin,t=pi-0.69473…+2pin

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sin(t)=(250)/(sqrt(250^2+300^2))

Solution

sin (t) = \frac{250}{25 0 ^{2} + 30 0 ^{2}}

Solution

t = 0.69473 \dots + 2 πn, t = π - 0.69473 \dots + 2 πn

Radians

t = 0.69473 \dots + 6.28318 \dots n, t = π - 0.69473 \dots + 6.28318 \dots n

Solution steps

sin (t) = \frac{250}{25 0 ^{2} + 30 0 ^{2}}

Simplify

\frac{250}{25 0 ^{2} + 30 0 ^{2}} : \frac{5 61}{61}

\frac{250}{25 0 ^{2} + 30 0 ^{2}}

25 0^{2} + 30 0^{2} = 152500

25 0^{2} + 30 0^{2}

25 0^{2} = 62500

= 62500 + 30 0^{2}

30 0^{2} = 90000

= 62500 + 90000

Add the numbers:

62500 + 90000 = 152500

= 152500

= \frac{250}{152500}

Factor

250 : 5^{3} \cdot 2

Factor

250 = 5^{3} \cdot 2

Factor

152500 : 5^{2} \cdot 261

Factor

152500 = 5^{4} \cdot 2^{2} \cdot 61

= 5^{4} \cdot 2^{2} \cdot 61

Apply radical rule:

= 61 2^{2} 5^{4}

Simplify

2^{2} : 2

2^{2}

Apply radical rule:

assuming

a \geq 0

= 2

Simplify

5^{4} : 5^{2}

5^{4}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

5^{4} = 5^{2 \cdot 2} = (5^{2})^{2}

= (5^{2})^{2}

Apply radical rule:

assuming

a \geq 0

= 5^{2}

= 5^{2} \cdot 261

= \frac{5 ^{3} \cdot 2}{5 ^{2} \cdot 2 61}

Cancel

\frac{2 \cdot 5 ^{3}}{2 \cdot 5 ^{2} 61} : \frac{5}{61}

\frac{2 \cdot 5 ^{3}}{2 \cdot 5 ^{2} 61}

Divide the numbers:

\frac{2}{2} = 1

= \frac{5 ^{3}}{5 ^{2} 61}

Apply exponent rule:

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

\frac{5 ^{3}}{5 ^{2}} = 5^{3 - 2}

= \frac{5 ^{3 - 2}}{61}

Subtract the numbers:

3 - 2 = 1

= \frac{5}{61}

= \frac{5}{61}

Rationalize

\frac{5}{61} : \frac{5 61}{61}

\frac{5}{61}

Multiply by the conjugate

\frac{61}{61}

= \frac{5 61}{61 61}

6161 = 61

6161

Apply radical rule:

a a = a

6161 = 61

= 61

= \frac{5 61}{61}

= \frac{5 61}{61}

sin (t) = \frac{5 61}{61}

Apply trig inverse properties

sin (t) = \frac{5 61}{61}

General solutions for

sin (t) = \frac{5 61}{61}

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

t = arcsin (\frac{5 61}{61}) + 2 πn, t = π - arcsin (\frac{5 61}{61}) + 2 πn

t = arcsin (\frac{5 61}{61}) + 2 πn, t = π - arcsin (\frac{5 61}{61}) + 2 πn

Show solutions in decimal form

t = 0.69473 \dots + 2 πn, t = π - 0.69473 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(t)=(250)/(sqrt(250^2+300^2)) ?
The general solution for sin(t)=(250)/(sqrt(250^2+300^2)) is t=0.69473…+2pin,t=pi-0.69473…+2pin