What is the value of 11sin(45) ?

The value of 11sin(45) is (11sqrt(2))/2

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11sin(45)

11 sin (4 5^{\circ})

\frac{11 2}{2}

Decimal Notation

7.77817 \dots

Solution steps

11 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}

= 11 \cdot \frac{2}{2}

Simplify

11 \cdot \frac{2}{2} : \frac{11 2}{2}

11 \cdot \frac{2}{2}

Multiply fractions:

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

= \frac{2 \cdot 11}{2}

Apply radical rule:

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

= \frac{11 \cdot 2 ^{\frac{1}{2}}}{2}

Apply exponent rule:

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

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

= \frac{11}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{11}{2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{11}{2}

Rationalize

\frac{11}{2} : \frac{11 2}{2}

\frac{11}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{11 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{11 2}{2}

= \frac{11 2}{2}

= \frac{11 2}{2}

3 sin (16 5^{\circ})

1 + sin (6 0^{\circ})

sin (3 0^{\circ} - 4 5^{\circ})

\frac{1}{4} (cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3}))

7.5 cos (\frac{2 π}{365} (31 - 7)) + 21.5