What is the value of 9/(sin(45)) ?

The value of 9/(sin(45)) is 9sqrt(2)

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9/(sin(45))

\frac{9}{sin ( 4 5 ^{\circ} )}

92

Decimal

12.72792 \dots

Solution steps

\frac{9}{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{9}{\frac{2}{2}}

Simplify

\frac{9}{\frac{2}{2}} : 92

\frac{9}{\frac{2}{2}}

Apply the fraction rule:

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

= \frac{9 \cdot 2}{2}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{18}{2}

Factor

18 : 3^{2} \cdot 2

Factor

18 = 3^{2} \cdot 2

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

Cancel

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

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

Apply radical rule:

n a = a^{\frac{1}{n}}

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply radical rule:

a^{\frac{1}{n}} = n a

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

= 3^{2} 2

= 2 \cdot 3^{2}

3^{2} = 9

= 92

= 92