What is the value of 8/(sin(45)) ?

The value of 8/(sin(45)) is 8sqrt(2)

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8/(sin(45))

\frac{8}{sin ( 4 5 ^{\circ} )}

82

Decimal Notation

11.31370 \dots

Solution steps

\frac{8}{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{8}{\frac{2}{2}}

Simplify

\frac{8}{\frac{2}{2}} : 82

\frac{8}{\frac{2}{2}}

Apply the fraction rule:

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

= \frac{8 \cdot 2}{2}

Multiply the numbers:

8 \cdot 2 = 16

= \frac{16}{2}

Factor

16 : 2^{4}

Factor

16 = 2^{4}

= \frac{2 ^{4}}{2}

Cancel

\frac{2 ^{4}}{2} : 2^{\frac{7}{2}}

\frac{2 ^{4}}{2}

Apply radical rule:

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

= \frac{2 ^{4}}{2 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{2 ^{4}}{2 ^{\frac{1}{2}}} = 2^{4 - \frac{1}{2}}

= 2^{4 - \frac{1}{2}}

Subtract the numbers:

4 - \frac{1}{2} = \frac{7}{2}

= 2^{\frac{7}{2}}

= 2^{\frac{7}{2}}

2^{\frac{7}{2}} = 2^{3} 2

2^{\frac{7}{2}}

2^{\frac{7}{2}} = 2^{3 + \frac{1}{2}}

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

Apply exponent rule:

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

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

Refine

= 2^{3} 2

= 2^{3} 2

2^{3} = 8

= 82

= 82

arctan (\frac{- 2}{1})

cos (0.5 π)

- arctan (\frac{1}{2})

\frac{cos ( 0 )}{1}

100 sin (2 5^{\circ})