What is the value of 50sin^3(pi/3) ?

The value of 50sin^3(pi/3) is (75sqrt(3))/4

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50sin^3(pi/3)

50 sin^{3} (\frac{π}{3})

\frac{75 3}{4}

Decimal Notation

32.47595 \dots

Solution steps

50 sin^{3} (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= 50 (\frac{3}{2})^{3}

Simplify

50 (\frac{3}{2})^{3} : \frac{75 3}{4}

50 (\frac{3}{2})^{3}

(\frac{3}{2})^{3} = \frac{3 3}{2 ^{3}}

(\frac{3}{2})^{3}

Apply exponent rule:

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

= \frac{( 3 ) ^{3}}{2 ^{3}}

(3)^{3} : 3^{\frac{3}{2}}

Apply radical rule:

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

= (3^{\frac{1}{2}})^{3}

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 3

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

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

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

3^{\frac{3}{2}} = 33

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

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

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

Apply exponent rule:

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

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

Refine

= 33

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

= 50 \cdot \frac{3 3}{2 ^{3}}

Multiply fractions:

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

= \frac{3 3 \cdot 50}{2 ^{3}}

Multiply the numbers:

3 \cdot 50 = 150

= \frac{150 3}{2 ^{3}}

Factor

150 : 5^{2} \cdot 2 \cdot 3

Factor

150 = 5^{2} \cdot 2 \cdot 3

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

Cancel the common factor:

2

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

5^{2} \cdot 33 = 753

5^{2} \cdot 33

5^{2} = 25

= 25 \cdot 33

Multiply the numbers:

25 \cdot 3 = 75

= 753

= \frac{75 3}{2 ^{2}}

2^{2} = 4

= \frac{75 3}{4}

= \frac{75 3}{4}

cos^{4} (3 0^{\circ})

cos (\frac{π}{6} + \frac{1}{4})

sin (7 5^{\circ}) sin (4 5^{\circ})

\frac{2}{tan ( 2 5 ^{\circ} )}

- sin (\frac{π}{2}) - cos (\frac{π}{2})

What is the value of 50sin^3(pi/3) ?
The value of 50sin^3(pi/3) is (75sqrt(3))/4