What is the value of -26/3 sin(3*2/3 pi)+3cos(3*2/3 pi)-2/9 ?

The value of -26/3 sin(3*2/3 pi)+3cos(3*2/3 pi)-2/9 is 25/9

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-26/3 sin(32/3 pi)+3cos(32/3 pi)-2/9

Solution

- \frac{26}{3} sin (3 \cdot \frac{2}{3} π) + 3 cos (3 \cdot \frac{2}{3} π) - \frac{2}{9}

Solution

\frac{25}{9}

Decimal Notation

2.77777 \dots

Solution steps

- \frac{26}{3} sin (3 \cdot \frac{2}{3} π) + 3 cos (3 \cdot \frac{2}{3} π) - \frac{2}{9}

Simplify:

3 \cdot \frac{2}{3} π = 2 π

3 \cdot \frac{2}{3} π

Multiply fractions:

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

= \frac{2 \cdot 3 π}{3}

Cancel the common factor:

3

= 2 π

- \frac{26}{3} sin (2 π) + 3 cos (2 π) - \frac{2}{9} = \frac{- 2 - 78 sin ( 2 π ) + 27 cos ( 2 π )}{9}

- \frac{26}{3} sin (2 π) + 3 cos (2 π) - \frac{2}{9}

Multiply

\frac{26}{3} sin (2 π) : \frac{26 sin ( 2 π )}{3}

\frac{26}{3} sin (2 π)

Multiply fractions:

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

= \frac{26 sin ( 2 π )}{3}

= - \frac{26 sin ( 2 π )}{3} + 3 cos (2 π) - \frac{2}{9}

Group like terms

= - \frac{2}{9} - \frac{26 sin ( 2 π )}{3} + 3 cos (2 π)

Convert element to fraction:

3 cos (2 π) = \frac{3 cos ( 2 π )}{1}

= - \frac{2}{9} - \frac{26 sin ( 2 π )}{3} + \frac{3 cos ( 2 π )}{1}

Least Common Multiplier of

9, 3, 1 : 9

9, 3, 1

Least Common Multiplier (LCM)

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

1

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

9, 3, 1

= 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{26 sin ( 2 π )}{3} :

multiply the denominator and numerator by

3

\frac{26 sin ( 2 π )}{3} = \frac{26 sin ( 2 π ) \cdot 3}{3 \cdot 3} = \frac{78 sin ( 2 π )}{9}

For

\frac{3 cos ( 2 π )}{1} :

multiply the denominator and numerator by

9

\frac{3 cos ( 2 π )}{1} = \frac{3 cos ( 2 π ) \cdot 9}{1 \cdot 9} = \frac{27 cos ( 2 π )}{9}

= - \frac{2}{9} - \frac{78 sin ( 2 π )}{9} + \frac{27 cos ( 2 π )}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 - 78 sin ( 2 π ) + 27 cos ( 2 π )}{9}

= \frac{- 2 - 78 sin ( 2 π ) + 27 cos ( 2 π )}{9}

sin (2 π) = sin (0)

sin (2 π)

Rewrite

2 π

2 π + 0

= sin (2 π + 0)

Apply the periodicity of

sin

sin (x + 2 π) = sin (x)

sin (2 π + 0) = sin (0)

= sin (0)

= \frac{- 2 - 78 sin ( 0 ) + 27 cos ( 2 π )}{9}

cos (2 π) = cos (0)

cos (2 π)

Rewrite

2 π

2 π + 0

= cos (2 π + 0)

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

cos (2 π + 0) = cos (0)

= cos (0)

= \frac{- 2 - 78 sin ( 0 ) + 27 cos ( 0 )}{9}

Use the following trivial identity:

sin (0) = 0

sin (0)

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

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

= \frac{- 2 - 78 \cdot 0 + 27 \cdot 1}{9}

Simplify

= \frac{25}{9}

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Frequently Asked Questions (FAQ)

What is the value of -26/3 sin(3*2/3 pi)+3cos(3*2/3 pi)-2/9 ?
The value of -26/3 sin(3*2/3 pi)+3cos(3*2/3 pi)-2/9 is 25/9