What is the value of -sec(pi/6)+2sec^3(pi/6) ?

The value of -sec(pi/6)+2sec^3(pi/6) is (10sqrt(3))/9

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

-sec(pi/6)+2sec^3(pi/6)

Solution

- sec (\frac{π}{6}) + 2 sec^{3} (\frac{π}{6})

Solution

\frac{10 3}{9}

Decimal Notation

1.92450 \dots

Solution steps

- sec (\frac{π}{6}) + 2 sec^{3} (\frac{π}{6})

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

Simplify

- \frac{2 3}{3} + 2 (\frac{2 3}{3})^{3} : \frac{10 3}{9}

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

2 (\frac{2 3}{3})^{3} = \frac{16 3}{9}

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

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

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

Apply exponent rule:

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

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

Apply exponent rule:

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

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

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

= 3^{\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}

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

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

3^{\frac{3}{2}} = 33

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

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

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 33

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

Cancel the common factor:

3

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

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

Multiply fractions:

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

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

2^{3} 3 \cdot 2 = 2^{4} 3

2^{3} 3 \cdot 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= 3 \cdot 2^{3 + 1}

Add the numbers:

3 + 1 = 4

= 3 \cdot 2^{4}

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

2^{4} = 16

= \frac{16 3}{3 ^{2}}

3^{2} = 9

= \frac{16 3}{9}

= - \frac{2 3}{3} + \frac{16 3}{9}

Least Common Multiplier of

3, 9 : 9

3, 9

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

3

9

= 3 \cdot 3

Multiply the numbers:

3 \cdot 3 = 9

= 9

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

9

For

\frac{2 3}{3} :

multiply the denominator and numerator by

3

\frac{2 3}{3} = \frac{2 3 \cdot 3}{3 \cdot 3} = \frac{6 3}{9}

= - \frac{6 3}{9} + \frac{3 \cdot 16}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{- 6 3 + 3 \cdot 16}{9}

Add similar elements:

- 63 + 163 = 103

= \frac{10 3}{9}

= \frac{10 3}{9}

Popular Examples

6cos((5pi)/3)

6 cos (\frac{5 π}{3})

sin(11.25)

sin (11.2 5^{\circ})

ln(sec(pi/6)+tan(pi/6))

ln (sec (\frac{π}{6}) + tan (\frac{π}{6}))

sin(355)

sin (35 5^{\circ})

1/(tan(2))

\frac{1}{tan ( 2 )}

Frequently Asked Questions (FAQ)

What is the value of -sec(pi/6)+2sec^3(pi/6) ?
The value of -sec(pi/6)+2sec^3(pi/6) is (10sqrt(3))/9