What is the value of sec^2(pi/6)-4 ?

The value of sec^2(pi/6)-4 is -8/3

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sec^2(pi/6)-4

sec^{2} (\frac{π}{6}) - 4

- \frac{8}{3}

Decimal Notation

- 2.66666 \dots

Solution steps

sec^{2} (\frac{π}{6}) - 4

Use the following trivial identity:

sec (\frac{π}{6}) = \frac{2 3}{3}

sec (\frac{π}{6})

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}

= \frac{2 3}{3}

= (\frac{2 3}{3})^{2} - 4

Simplify

(\frac{2 3}{3})^{2} - 4 : - \frac{8}{3}

(\frac{2 3}{3})^{2} - 4

(\frac{2 3}{3})^{2} = \frac{2 ^{2}}{3}

(\frac{2 3}{3})^{2}

Apply exponent rule:

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

= \frac{( 2 3 ) ^{2}}{3 ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(23)^{2} = 2^{2} (3)^{2}

= \frac{2 ^{2} ( 3 ) ^{2}}{3 ^{2}}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

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

Cancel the common factor:

3

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

= \frac{2 ^{2}}{3} - 4

2^{2} = 4

= \frac{4}{3} - 4

Convert element to fraction:

4 = \frac{4 \cdot 3}{3}

= - \frac{4 \cdot 3}{3} + \frac{4}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 \cdot 3 + 4}{3}

- 4 \cdot 3 + 4 = - 8

- 4 \cdot 3 + 4

Multiply the numbers:

4 \cdot 3 = 12

= - 12 + 4

Add/Subtract the numbers:

- 12 + 4 = - 8

= - 8

= \frac{- 8}{3}

Apply the fraction rule:

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

= - \frac{8}{3}

= - \frac{8}{3}

sin (2) (\frac{12}{13})

\frac{6 tan ( \frac{4 π}{7} )}{tan ^{2} ( \frac{4 π}{7} ) - 1}

cos (\frac{7 π}{15})

cos (\frac{11 π}{12}) cos (\frac{3 π}{4}) + sin (\frac{11 π}{12}) sin (\frac{3 π}{4})

9 sin (\frac{π}{2}) + 9 cos (\frac{π}{2})