What is the value of 1/3 (-8cos(pi/3)-(tan^2(pi/3))/2) ?

The value of 1/3 (-8cos(pi/3)-(tan^2(pi/3))/2) is -11/6

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1/3 (-8cos(pi/3)-(tan^2(pi/3))/2)

Solution

\frac{1}{3} (- 8 cos (\frac{π}{3}) - \frac{tan ^{2} ( \frac{π}{3} )}{2})

Solution

- \frac{11}{6}

Decimal Notation

- 1.83333 \dots

Solution steps

\frac{1}{3} (- 8 cos (\frac{π}{3}) - \frac{tan ^{2} ( \frac{π}{3} )}{2})

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}

Use the following trivial identity:

tan (\frac{π}{3}) = 3

tan (\frac{π}{3})

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 3

= \frac{1}{3} (- 8 \cdot \frac{1}{2} - \frac{( 3 ) ^{2}}{2})

Simplify

\frac{1}{3} (- 8 \cdot \frac{1}{2} - \frac{( 3 ) ^{2}}{2}) : - \frac{11}{6}

\frac{1}{3} (- 8 \cdot \frac{1}{2} - \frac{( 3 ) ^{2}}{2})

8 \cdot \frac{1}{2} = 4

8 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 8}{2}

Multiply the numbers:

1 \cdot 8 = 8

= \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4

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

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

(3)^{2} = 3

(3)^{2}

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{3}{2}

= \frac{1}{3} (- \frac{3}{2} - 4)

Join

- 4 - \frac{3}{2} : - \frac{11}{2}

- 4 - \frac{3}{2}

Convert element to fraction:

4 = \frac{4 \cdot 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

- 4 \cdot 2 - 3 = - 11

- 4 \cdot 2 - 3

Multiply the numbers:

4 \cdot 2 = 8

= - 8 - 3

Subtract the numbers:

- 8 - 3 = - 11

= - 11

= \frac{- 11}{2}

Apply the fraction rule:

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

= - \frac{11}{2}

= \frac{1}{3} (- \frac{11}{2})

Remove parentheses:

(- a) = - a

= - \frac{1}{3} \cdot \frac{11}{2}

Multiply fractions:

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

= - \frac{1 \cdot 11}{3 \cdot 2}

Multiply the numbers:

1 \cdot 11 = 11

= - \frac{11}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= - \frac{11}{6}

= - \frac{11}{6}

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Frequently Asked Questions (FAQ)

What is the value of 1/3 (-8cos(pi/3)-(tan^2(pi/3))/2) ?
The value of 1/3 (-8cos(pi/3)-(tan^2(pi/3))/2) is -11/6