What is the value of 1.5sin(524*pi*4.45) ?

The value of 1.5sin(524*pi*4.45) is -(3sqrt(2)sqrt(5-\sqrt{5)})/8

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1.5sin(524pi4.45)

Solution

1.5 sin (524 \cdot π \cdot 4.45)

Solution

- \frac{3 2 5 - 5}{8}

Decimal Notation

- 0.88167 \dots

Solution steps

1.5 sin (524 π 4.45)

= \frac{3}{2} sin (524 π \frac{89}{20})

Simplify:

524 π \frac{89}{20} = \frac{11659 π}{5}

524 π \frac{89}{20}

Multiply fractions:

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

= \frac{89 \cdot 524 π}{20}

Multiply the numbers:

89 \cdot 524 = 46636

= \frac{46636 π}{20}

Cancel the common factor:

4

= \frac{11659 π}{5}

\frac{3}{2} sin (\frac{11659 π}{5}) = \frac{3 sin ( \frac{11659 π}{5} )}{2}

\frac{3}{2} sin (\frac{11659 π}{5})

Multiply fractions:

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

= \frac{3 sin ( \frac{11659 π}{5} )}{2}

= \frac{3 sin ( \frac{11659 π}{5} )}{2}

sin (\frac{11659 π}{5}) = sin (\frac{9 π}{5})

sin (\frac{11659 π}{5})

Rewrite

\frac{11659 π}{5}

2 π \cdot 1165 + \frac{9 π}{5}

= sin (2 π 1165 + \frac{9 π}{5})

Apply the periodicity of

sin

sin (x + 2 π \cdot k) = sin (x)

sin (2 π \cdot 1165 + \frac{9 π}{5}) = sin (\frac{9 π}{5})

= sin (\frac{9 π}{5})

= \frac{3 sin ( \frac{9 π}{5} )}{2}

Rewrite using trig identities:

sin (\frac{9 π}{5}) = - \frac{2 5 - 5}{4}

sin (\frac{9 π}{5})

Rewrite using trig identities:

- sin (\frac{π}{5})

sin (\frac{9 π}{5})

Use the following identity:

sin (x) = - sin (2 π - x)

sin (x)

Use the following property:

sin (θ) = - sin (- θ)

sin (x) = - sin (- x)

= - sin (- x)

Apply the periodicity of

sin

sin (2 π + θ) = sin (θ)

- sin (- x) = - sin (2 π - x)

= - sin (2 π - x)

= - sin (2 π - \frac{9 π}{5})

Simplify:

2 π - \frac{9 π}{5} = \frac{π}{5}

2 π - \frac{9 π}{5}

Convert element to fraction:

2 π = \frac{2 π 5}{5}

= \frac{2 π 5}{5} - \frac{9 π}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 5 - 9 π}{5}

2 π 5 - 9 π = π

2 π 5 - 9 π

Multiply the numbers:

2 \cdot 5 = 10

= 10 π - 9 π

Add similar elements:

10 π - 9 π = π

= π

= \frac{π}{5}

= - sin (\frac{π}{5})

= - sin (\frac{π}{5})

Rewrite using trig identities:

sin (\frac{π}{5}) = \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}

= \frac{3 ( - \frac{2 5 - 5}{4} )}{2}

Simplify

\frac{3 ( - \frac{2 5 - 5}{4} )}{2} : - \frac{3 2 5 - 5}{8}

\frac{3 ( - \frac{2 5 - 5}{4} )}{2}

Remove parentheses:

(- a) = - a

= \frac{- 3 \cdot \frac{2 5 - 5}{4}}{2}

Apply the fraction rule:

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

= - \frac{3 \cdot \frac{2 5 - 5}{4}}{2}

Multiply

3 \cdot \frac{2 5 - 5}{4} : \frac{3 5 - 5}{2 2}

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

Multiply fractions:

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

= \frac{2 5 - 5 \cdot 3}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{3 2 5 - 5}{2 ^{2}}

Cancel

\frac{2 5 - 5 \cdot 3}{2 ^{2}} : \frac{3 5 - 5}{2 ^{\frac{3}{2}}}

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

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

= \frac{3 5 - 5}{2 ^{\frac{3}{2}}}

= \frac{3 5 - 5}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

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

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

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

Apply exponent rule:

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

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

Refine

= 22

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

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

Simplify

\frac{\frac{3 5 - 5}{2 2}}{2} : \frac{3 5 - 5}{4 2}

\frac{\frac{3 5 - 5}{2 2}}{2}

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 5 - 5}{4 2}

= - \frac{3 5 - 5}{4 2}

Rationalize

- \frac{3 5 - 5}{4 2} : - \frac{3 2 5 - 5}{8}

- \frac{3 5 - 5}{4 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{3 5 - 5 2}{4 2 2}

422 = 8

422

Apply radical rule:

a a = a

22 = 2

= 4 \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= 8

= - \frac{3 2 5 - 5}{8}

= - \frac{3 2 5 - 5}{8}

= - \frac{3 2 5 - 5}{8}

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Frequently Asked Questions (FAQ)

What is the value of 1.5sin(524*pi*4.45) ?
The value of 1.5sin(524*pi*4.45) is -(3sqrt(2)sqrt(5-\sqrt{5)})/8