What is the value of -1/(cos(pi/4)) ?

The value of -1/(cos(pi/4)) is -sqrt(2)

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-1/(cos(pi/4))

- \frac{1}{cos ( \frac{π}{4} )}

- 2

Decimal Notation

- 1.41421 \dots

Solution steps

- \frac{1}{cos ( \frac{π}{4} )}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= - \frac{1}{\frac{2}{2}}

Simplify

- \frac{1}{\frac{2}{2}} : - 2

- \frac{1}{\frac{2}{2}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= - \frac{2}{2}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

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

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

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

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

Apply radical rule:

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

= - 2

= - 2

sin (2 \cdot 2 π)

sin (3 5^{\circ}) cos (2 5^{\circ}) + cos (3 5^{\circ}) sin (2 5^{\circ})

arctan (53)

cot (\frac{9 π}{2})

arctan (- \frac{π}{6})