What is the value of (90)/(sin(45)) ?

The value of (90)/(sin(45)) is 90sqrt(2)

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(90)/(sin(45))

\frac{90}{sin ( 4 5 ^{\circ} )}

902

Decimal

127.27922 \dots

Solution steps

\frac{90}{sin ( 4 5 ^{\circ} )}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{90}{\frac{2}{2}}

Simplify

\frac{90}{\frac{2}{2}} : 902

\frac{90}{\frac{2}{2}}

Apply the fraction rule:

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

= \frac{90 \cdot 2}{2}

Multiply the numbers:

90 \cdot 2 = 180

= \frac{180}{2}

Factor

180 : 2^{2} \cdot 3^{2} \cdot 5

Factor

180 = 2^{2} \cdot 3^{2} \cdot 5

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

Cancel

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

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

Apply radical rule:

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

= \frac{2 ^{2} \cdot 3 ^{2} \cdot 5}{2 ^{\frac{1}{2}}}

Apply exponent rule:

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

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

= 3^{2} \cdot 5 \cdot 2^{- \frac{1}{2} + 2}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

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

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

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= 3^{2} \cdot 5 \cdot 22

Multiply the numbers:

5 \cdot 2 = 10

= 3^{2} \cdot 102

3^{2} = 9

= 10 \cdot 92

Multiply the numbers:

10 \cdot 9 = 90

= 902

= 902