What is the value of 40pi[0.6+((sin(8pi*0.6))/(8pi))] ?

The value of 40pi[0.6+((sin(8pi*0.6))/(8pi))] is 24pi+(5sqrt(2)sqrt(5-\sqrt{5)})/4

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40pi[0.6+((sin(8pi*0.6))/(8pi))]

Solution

40 π [0.6 + (\frac{sin ( 8 π \cdot 0.6 )}{8 π})]

Solution

24 π + \frac{5 2 5 - 5}{4}

Decimal Notation

78.33714 \dots

Solution steps

40 π [0.6 + (\frac{sin ( 8 π 0.6 )}{8 π})]

= 40 π [\frac{3}{5} + \frac{sin ( 8 π \frac{3}{5} )}{8 π}]

Simplify:

8 π \frac{3}{5} = \frac{24 π}{5}

8 π \frac{3}{5}

Multiply fractions:

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

= \frac{3 \cdot 8 π}{5}

Multiply the numbers:

3 \cdot 8 = 24

= \frac{24 π}{5}

40 π [\frac{3}{5} + \frac{sin ( \frac{24 π}{5} )}{8 π}] = 24 π + 5 sin (\frac{24 π}{5})

40 π (\frac{3}{5} + \frac{sin ( \frac{24 π}{5} )}{8 π})

Join

\frac{3}{5} + \frac{sin ( \frac{24 π}{5} )}{8 π} : \frac{24 π + 5 sin ( \frac{24 π}{5} )}{40 π}

\frac{3}{5} + \frac{sin ( \frac{24 π}{5} )}{8 π}

Least Common Multiplier of

5, 8 π : 40 π

5, 8 π

Lowest Common Multiplier (LCM)

Least Common Multiplier of

5, 8 : 40

5, 8

Least Common Multiplier (LCM)

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

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

5

8

= 5 \cdot 2 \cdot 2 \cdot 2

Multiply the numbers:

5 \cdot 2 \cdot 2 \cdot 2 = 40

= 40

Compute an expression comprised of factors that appear either in

5

8 π

= 40 π

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

40 π

For

\frac{3}{5} :

multiply the denominator and numerator by

8 π

\frac{3}{5} = \frac{3 \cdot 8 π}{5 \cdot 8 π} = \frac{24 π}{40 π}

For

\frac{sin ( \frac{24 π}{5} )}{8 π} :

multiply the denominator and numerator by

5

\frac{sin ( \frac{24 π}{5} )}{8 π} = \frac{sin ( \frac{24 π}{5} ) \cdot 5}{8 π 5} = \frac{sin ( \frac{24 π}{5} ) \cdot 5}{40 π}

= \frac{24 π}{40 π} + \frac{sin ( \frac{24 π}{5} ) \cdot 5}{40 π}

Since the denominators are equal, combine the fractions:

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

= \frac{24 π + sin ( \frac{24 π}{5} ) \cdot 5}{40 π}

= 40 π \frac{24 π + 5 sin ( \frac{24 π}{5} )}{40 π}

Multiply fractions:

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

= \frac{( 24 π + sin ( \frac{24 π}{5} ) \cdot 5 ) \cdot 40 π}{40 π}

Cancel the common factor:

40

= \frac{( 24 π + sin ( \frac{24 π}{5} ) \cdot 5 ) π}{π}

Cancel the common factor:

π

= 24 π + sin (\frac{24 π}{5}) \cdot 5

= 24 π + 5 sin (\frac{24 π}{5})

sin (\frac{24 π}{5}) = sin (\frac{4 π}{5})

sin (\frac{24 π}{5})

Rewrite

\frac{24 π}{5}

2 π \cdot 2 + \frac{4 π}{5}

= sin (2 π 2 + \frac{4 π}{5})

Apply the periodicity of

sin

sin (x + 2 π \cdot k) = sin (x)

sin (2 π \cdot 2 + \frac{4 π}{5}) = sin (\frac{4 π}{5})

= sin (\frac{4 π}{5})

= 24 π + 5 sin (\frac{4 π}{5})

Rewrite using trig identities:

sin (\frac{4 π}{5}) = \frac{2 5 - 5}{4}

sin (\frac{4 π}{5})

Rewrite using trig identities:

sin (\frac{π}{5})

sin (\frac{4 π}{5})

Use the basic trigonometric identity:

sin (x) = sin (π - x)

= sin (π - \frac{4 π}{5})

Simplify:

π - \frac{4 π}{5} = \frac{π}{5}

π - \frac{4 π}{5}

Convert element to fraction:

π = \frac{π 5}{5}

= \frac{π 5}{5} - \frac{4 π}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{π 5 - 4 π}{5}

Add similar elements:

5 π - 4 π = π

= \frac{π}{5}

= sin (\frac{π}{5})

= sin (\frac{π}{5})

Rewrite using trig identities:

\frac{2 5 - 5}{4}

sin (\frac{π}{5})

Show that:

cos (\frac{π}{5}) - sin (\frac{π}{10}) = \frac{1}{2}

Use the following product to sum identity:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (\frac{π}{5}) sin (\frac{π}{10}) = sin (\frac{3 π}{10}) - sin (\frac{π}{10})

Show that:

2 cos (\frac{π}{5}) sin (\frac{π}{10}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (\frac{2 π}{5}) = 2 sin (\frac{π}{5}) cos (\frac{π}{5})

sin (\frac{2 π}{5}) sin (\frac{π}{5}) = 4 sin (\frac{π}{5}) sin (\frac{π}{10}) cos (\frac{π}{5}) cos (\frac{π}{10})

Divide both sides by

sin (\frac{π}{5})

sin (\frac{2 π}{5}) = 4 sin (\frac{π}{10}) cos (\frac{π}{5}) cos (\frac{π}{10})

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

sin (\frac{2 π}{5}) = cos (\frac{π}{2} - \frac{2 π}{5})

cos (\frac{π}{2} - \frac{2 π}{5}) = 4 sin (\frac{π}{10}) cos (\frac{π}{5}) cos (\frac{π}{10})

cos (\frac{π}{10}) = 4 sin (\frac{π}{10}) cos (\frac{π}{5}) cos (\frac{π}{10})

Divide both sides by

cos (\frac{π}{10})

1 = 4 sin (\frac{π}{10}) cos (\frac{π}{5})

Divide both sides by

2

\frac{1}{2} = 2 sin (\frac{π}{10}) cos (\frac{π}{5})

Substitute

\frac{1}{2} = 2 sin (\frac{π}{10}) cos (\frac{π}{5})

\frac{1}{2} = sin (\frac{3 π}{10}) - sin (\frac{π}{10})

sin (\frac{3 π}{10}) = cos (\frac{π}{2} - \frac{3 π}{10})

\frac{1}{2} = cos (\frac{π}{2} - \frac{3 π}{10}) - sin (\frac{π}{10})

\frac{1}{2} = cos (\frac{π}{5}) - sin (\frac{π}{10})

Show that:

cos (\frac{π}{5}) + sin (\frac{π}{10}) = \frac{5}{4}

Use the factorization rule:

a^{2} - b^{2} = (a + b) (a - b)

a = cos (\frac{π}{5}) + sin (\frac{π}{10})

(cos (\frac{π}{5}) + sin (\frac{π}{10}))^{2} - (cos (\frac{π}{5}) - sin (\frac{π}{10}))^{2} = ((cos (\frac{π}{5}) + sin (\frac{π}{10})) + (cos (\frac{π}{5}) - sin (\frac{π}{10}))) ((cos (\frac{π}{5}) + sin (\frac{π}{10})) - (cos (\frac{π}{5}) - sin (\frac{π}{10})))

Refine

(cos (\frac{π}{5}) + sin (\frac{π}{10}))^{2} - (cos (\frac{π}{5}) - sin (\frac{π}{10}))^{2} = 2 (2 cos (\frac{π}{5}) sin (\frac{π}{10}))

Show that:

2 cos (\frac{π}{5}) sin (\frac{π}{10}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (\frac{2 π}{5}) = 2 sin (\frac{π}{5}) cos (\frac{π}{5})

sin (\frac{2 π}{5}) sin (\frac{π}{5}) = 4 sin (\frac{π}{5}) sin (\frac{π}{10}) cos (\frac{π}{5}) cos (\frac{π}{10})

Divide both sides by

sin (\frac{π}{5})

sin (\frac{2 π}{5}) = 4 sin (\frac{π}{10}) cos (\frac{π}{5}) cos (\frac{π}{10})

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

sin (\frac{2 π}{5}) = cos (\frac{π}{2} - \frac{2 π}{5})

cos (\frac{π}{2} - \frac{2 π}{5}) = 4 sin (\frac{π}{10}) cos (\frac{π}{5}) cos (\frac{π}{10})

cos (\frac{π}{10}) = 4 sin (\frac{π}{10}) cos (\frac{π}{5}) cos (\frac{π}{10})

Divide both sides by

cos (\frac{π}{10})

1 = 4 sin (\frac{π}{10}) cos (\frac{π}{5})

Divide both sides by

2

\frac{1}{2} = 2 sin (\frac{π}{10}) cos (\frac{π}{5})

Substitute

2 cos (\frac{π}{5}) sin (\frac{π}{10}) = \frac{1}{2}

(cos (\frac{π}{5}) + sin (\frac{π}{10}))^{2} - (cos (\frac{π}{5}) - sin (\frac{π}{10}))^{2} = 1

Substitute

cos (\frac{π}{5}) - sin (\frac{π}{10}) = \frac{1}{2}

(cos (\frac{π}{5}) + sin (\frac{π}{10}))^{2} - (\frac{1}{2})^{2} = 1

Refine

(cos (\frac{π}{5}) + sin (\frac{π}{10}))^{2} - \frac{1}{4} = 1

Add

\frac{1}{4}

to both sides

(cos (\frac{π}{5}) + sin (\frac{π}{10}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

Refine

(cos (\frac{π}{5}) + sin (\frac{π}{10}))^{2} = \frac{5}{4}

Take the square root of both sides

cos (\frac{π}{5}) + sin (\frac{π}{10}) = \pm \frac{5}{4}

cos (\frac{π}{5})

cannot be negative

sin (\frac{π}{10})

cannot be negative

cos (\frac{π}{5}) + sin (\frac{π}{10}) = \frac{5}{4}

Add the following equations

cos (\frac{π}{5}) + sin (\frac{π}{10}) = \frac{5}{2}

((cos (\frac{π}{5}) + sin (\frac{π}{10})) + (cos (\frac{π}{5}) - sin (\frac{π}{10}))) = (\frac{5}{2} + \frac{1}{2})

Refine

cos (\frac{π}{5}) = \frac{5 + 1}{4}

Square both sides

(cos (\frac{π}{5}))^{2} = (\frac{5 + 1}{4})^{2}

Use the following identity:

sin^{2} (x) = 1 - cos^{2} (x)

sin^{2} (\frac{π}{5}) = 1 - cos^{2} (\frac{π}{5})

Substitute

cos (\frac{π}{5}) = \frac{5 + 1}{4}

sin^{2} (\frac{π}{5}) = 1 - (\frac{5 + 1}{4})^{2}

Refine

sin^{2} (\frac{π}{5}) = \frac{5 - 5}{8}

Take the square root of both sides

sin (\frac{π}{5}) = \pm \frac{5 - 5}{8}

sin (\frac{π}{5})

cannot be negative

sin (\frac{π}{5}) = \frac{5 - 5}{8}

Refine

sin (\frac{π}{5}) = \frac{\frac{5 - 5}{2}}{2}

= \frac{\frac{5 - 5}{2}}{2}

\frac{\frac{5 - 5}{2}}{2} = \frac{2 5 - 5}{4}

\frac{\frac{5 - 5}{2}}{2}

\frac{5 - 5}{2} = \frac{5 - 5}{2}

\frac{5 - 5}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5 - 5}{2}

= \frac{\frac{5 - 5}{2}}{2}

Apply the fraction rule:

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

= \frac{5 - 5}{2 \cdot 2}

Rationalize

\frac{5 - 5}{2 2} : \frac{2 5 - 5}{4}

\frac{5 - 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{5 - 5 2}{2 \cdot 2 2}

2 \cdot 22 = 4

2 \cdot 22

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= 24 π + 5 \cdot \frac{2 5 - 5}{4}

Simplify

24 π + 5 \cdot \frac{2 5 - 5}{4} : 24 π + \frac{5 2 5 - 5}{4}

24 π + 5 \cdot \frac{2 5 - 5}{4}

Multiply

5 \cdot \frac{2 5 - 5}{4} : \frac{5 2 5 - 5}{4}

5 \cdot \frac{2 5 - 5}{4}

Multiply fractions:

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

= \frac{2 5 - 5 \cdot 5}{4}

= 24 π + \frac{5 2 5 - 5}{4}

= 24 π + \frac{5 2 5 - 5}{4}

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Frequently Asked Questions (FAQ)

What is the value of 40pi[0.6+((sin(8pi*0.6))/(8pi))] ?
The value of 40pi[0.6+((sin(8pi*0.6))/(8pi))] is 24pi+(5sqrt(2)sqrt(5-\sqrt{5)})/4