What is the value of -sec(pi/4)+2sec^3(pi/4) ?

The value of -sec(pi/4)+2sec^3(pi/4) is 3sqrt(2)

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-sec(pi/4)+2sec^3(pi/4)

- sec (\frac{π}{4}) + 2 sec^{3} (\frac{π}{4})

32

Decimal Notation

4.24264 \dots

Solution steps

- sec (\frac{π}{4}) + 2 sec^{3} (\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 + 2 (2)^{3}

Simplify

- 2 + 2 (2)^{3} : 32

- 2 + 2 (2)^{3}

2 (2)^{3} = 42

2 (2)^{3}

(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

= 2 \cdot 22

Multiply the numbers:

2 \cdot 2 = 4

= 42

= - 2 + 42

Add similar elements:

- 2 + 42 = 32

= 32

= 32

arcsin (0.85)

arcsin (0.65)

arcsin (0.68)

arcsin (0.61)

sin (\frac{8}{17})

What is the value of -sec(pi/4)+2sec^3(pi/4) ?
The value of -sec(pi/4)+2sec^3(pi/4) is 3sqrt(2)