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

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

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csc^3(pi/4)

csc^{3} (\frac{π}{4})

22

Decimal Notation

2.82842 \dots

Solution steps

csc^{3} (\frac{π}{4})

Use the following trivial identity:

csc (\frac{π}{4}) = 2

csc (\frac{π}{4})

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= 2

= (2)^{3}

Simplify

(2)^{3} : 22

(2)^{3}

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}}

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

= 22

= 22

sin (- 78 0^{\circ})

cot (tan (\frac{21}{20}))

tan (2) (π)

2 (cos (4 5^{\circ}) + i sin (4 5^{\circ}))

4 (sin (4) + cos (4) + 4 (cos (4) - sin (4)))