What is the value of -cos(0)+(2cos^3(0))/3-(cos^5(0))/5 ?

The value of -cos(0)+(2cos^3(0))/3-(cos^5(0))/5 is -8/15

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Popular Trigonometry >

-cos(0)+(2cos^3(0))/3-(cos^5(0))/5

Solution

- cos (0) + \frac{2 cos ^{3} ( 0 )}{3} - \frac{cos ^{5} ( 0 )}{5}

Solution

- \frac{8}{15}

Decimal

- 0.53333 \dots

Solution steps

- cos (0) + \frac{2 cos ^{3} ( 0 )}{3} - \frac{cos ^{5} ( 0 )}{5}

Use the following trivial identity:

cos (0) = 1

cos (0)

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}

= 1

= - 1 + \frac{2 \cdot 1 ^{3}}{3} - \frac{1 ^{5}}{5}

Simplify

- 1 + \frac{2 \cdot 1 ^{3}}{3} - \frac{1 ^{5}}{5} : - \frac{8}{15}

- 1 + \frac{2 \cdot 1 ^{3}}{3} - \frac{1 ^{5}}{5}

Apply rule

1^{a} = 1

1^{3} = 1, 1^{5} = 1

= - 1 + \frac{2 \cdot 1}{3} - \frac{1}{5}

Multiply the numbers:

2 \cdot 1 = 2

= - 1 + \frac{2}{3} - \frac{1}{5}

Convert element to fraction:

1 = \frac{1}{1}

= - \frac{1}{1} + \frac{2}{3} - \frac{1}{5}

Least Common Multiplier of

1, 3, 5 : 15

1, 3, 5

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 5

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

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

15

For

\frac{1}{1} :

multiply the denominator and numerator by

15

\frac{1}{1} = \frac{1 \cdot 15}{1 \cdot 15} = \frac{15}{15}

For

\frac{2}{3} :

multiply the denominator and numerator by

5

\frac{2}{3} = \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15}

For

\frac{1}{5} :

multiply the denominator and numerator by

3

\frac{1}{5} = \frac{1 \cdot 3}{5 \cdot 3} = \frac{3}{15}

= - \frac{15}{15} + \frac{10}{15} - \frac{3}{15}

Since the denominators are equal, combine the fractions:

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

= \frac{- 15 + 10 - 3}{15}

Add/Subtract the numbers:

- 15 + 10 - 3 = - 8

= \frac{- 8}{15}

Apply the fraction rule:

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

= - \frac{8}{15}

= - \frac{8}{15}

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Frequently Asked Questions (FAQ)

What is the value of -cos(0)+(2cos^3(0))/3-(cos^5(0))/5 ?
The value of -cos(0)+(2cos^3(0))/3-(cos^5(0))/5 is -8/15