What is the value of sec^5(pi/4) ?

The value of sec^5(pi/4) is 4sqrt(2)

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sec^5(pi/4)

sec^{5} (\frac{π}{4})

42

Decimal Notation

5.65685 \dots

Solution steps

sec^{5} (\frac{π}{4})

Use the following trivial identity:

sec (\frac{π}{4}) = 2

sec (\frac{π}{4})

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 2

= (2)^{5}

Simplify

(2)^{5} : 42

(2)^{5}

Apply radical rule:

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

= (2^{\frac{1}{2}})^{5}

Apply exponent rule:

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

= 2^{\frac{1}{2} \cdot 5}

\frac{1}{2} \cdot 5 = \frac{5}{2}

\frac{1}{2} \cdot 5

Multiply fractions:

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

= \frac{1 \cdot 5}{2}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{2}

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

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

2^{\frac{5}{2}}

2^{\frac{5}{2}} = 2^{2 + \frac{1}{2}}

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

Apply exponent rule:

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

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

Refine

= 2^{2} 2

= 2^{2} 2

2^{2} = 4

= 42

= 42

sin (\frac{2}{2})

sin (0.94)

- cos (- 3 π)

sin (- π) + cos (- π)

12 sin (3 2^{\circ})