What is the value of sin^3(pi/4) ?

The value of sin^3(pi/4) is (sqrt(2))/4

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sin^3(pi/4)

sin^{3} (\frac{π}{4})

\frac{2}{4}

Decimal Notation

0.35355 \dots

Solution steps

sin^{3} (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

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{2}{2}

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

Simplify

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

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

Apply exponent rule:

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

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

= 2^{\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}

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

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

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

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

Cancel the common factor:

2

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

2^{2} = 4

= \frac{2}{4}

= \frac{2}{4}

arcsin (\frac{8}{17})

sin (\frac{- 5 π}{6})

cos (arctan (- 1))

sin (arctan (- \frac{3}{3}))

sec (- \frac{4 π}{3})