What is the value of 1/(cos^2(pi/3)) ?

The value of 1/(cos^2(pi/3)) is 4

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1/(cos^2(pi/3))

\frac{1}{cos ^{2} ( \frac{π}{3} )}

4

Solution steps

\frac{1}{cos ^{2} ( \frac{π}{3} )}

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

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{1}{2}

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

Simplify

\frac{1}{( \frac{1}{2} ) ^{2}} : 4

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= 2^{2}

2^{2} = 4

= 4

= 4

arccos (\frac{5}{9 14})

cos (- 5 0^{\circ})

arcsin (\frac{1}{1.7606})

cos (32.0 1^{\circ})

sin (3 0^{2})