What is the value of 3/(cos(45)) ?

The value of 3/(cos(45)) is 3sqrt(2)

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3/(cos(45))

\frac{3}{cos ( 4 5 ^{\circ} )}

32

Decimal Notation

4.24264 \dots

Solution steps

\frac{3}{cos ( 4 5 ^{\circ} )}

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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{3}{\frac{2}{2}}

Simplify

\frac{3}{\frac{2}{2}} : 32

\frac{3}{\frac{2}{2}}

Apply the fraction rule:

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

= \frac{3 \cdot 2}{2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6}{2}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

= \frac{2 \cdot 3}{2}

Cancel

\frac{2 \cdot 3}{2} : 2 \cdot 3

\frac{2 \cdot 3}{2}

Apply radical rule:

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

= \frac{2 \cdot 3}{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}}

= 3 \cdot 2^{- \frac{1}{2} + 1}

Subtract the numbers:

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

= 3 \cdot 2^{\frac{1}{2}}

Apply radical rule:

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

= 32

= 2 \cdot 3

= 32

6 cos (- \frac{π}{4})

\frac{tan ( 7 3 ^{\circ} ) - tan ( 1 3 ^{\circ} )}{1 + tan ( 7 3 ^{\circ} ) tan ( 1 3 ^{\circ} )}

400 sin (4 5^{\circ})

arccos (\frac{3}{34})

\frac{2 tan ( \frac{2 π}{7} )}{1 - tan ^{2} ( \frac{2 π}{7} )}