What is the value of ((2.68*9.80665))/(sin(75)) ?

The value of ((2.68*9.80665))/(sin(75)) is (13140911(sqrt(6)-sqrt(2)))/(500000)

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((2.68*9.80665))/(sin(75))

Solution

\frac{( 2.68 \cdot 9.80665 )}{sin ( 7 5 ^{\circ} )}

Solution

\frac{13140911 ( 6 - 2 )}{500000}

Decimal Notation

27.20894 \dots

Solution steps

\frac{( 2.68 \cdot 9.80665 )}{sin ( 7 5 ^{\circ} )}

= \frac{\frac{67}{25} \cdot \frac{196133}{20000}}{sin ( 7 5 ^{\circ} )}

Rewrite using trig identities:

sin (7 5^{\circ}) = \frac{6 + 2}{4}

sin (7 5^{\circ})

Rewrite using trig identities:

sin (4 5^{\circ}) cos (3 0^{\circ}) + cos (4 5^{\circ}) sin (3 0^{\circ})

sin (7 5^{\circ})

Write

sin (7 5^{\circ})

sin (4 5^{\circ} + 3 0^{\circ})

= sin (4 5^{\circ} + 3 0^{\circ})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (4 5^{\circ}) cos (3 0^{\circ}) + cos (4 5^{\circ}) sin (3 0^{\circ})

= sin (4 5^{\circ}) cos (3 0^{\circ}) + cos (4 5^{\circ}) sin (3 0^{\circ})

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

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

Simplify

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} : \frac{6 + 2}{4}

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2}

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 3}{4}

Simplify

23 : 6

23

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= \frac{6}{4}

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

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

Multiply fractions:

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

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

Multiply:

2 \cdot 1 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

= \frac{6}{4} + \frac{2}{4}

Apply rule

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

= \frac{6 + 2}{4}

= \frac{6 + 2}{4}

= \frac{\frac{67}{25} \cdot \frac{196133}{20000}}{\frac{6 + 2}{4}}

Simplify

\frac{\frac{67}{25} \cdot \frac{196133}{20000}}{\frac{6 + 2}{4}} : \frac{13140911 ( 6 - 2 )}{500000}

\frac{\frac{67}{25} \cdot \frac{196133}{20000}}{\frac{6 + 2}{4}}

Apply the fraction rule:

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

= \frac{\frac{67}{25} \cdot \frac{196133}{20000} \cdot 4}{6 + 2}

Multiply

\frac{67}{25} \cdot \frac{196133}{20000} \cdot 4 : \frac{13140911}{125000}

\frac{67}{25} \cdot \frac{196133}{20000} \cdot 4

Multiply fractions:

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

= \frac{67 \cdot 196133 \cdot 4}{25 \cdot 20000}

Multiply the numbers:

67 \cdot 196133 \cdot 4 = 52563644

= \frac{52563644}{25 \cdot 20000}

Multiply the numbers:

25 \cdot 20000 = 500000

= \frac{52563644}{500000}

Cancel the common factor:

4

= \frac{13140911}{125000}

= \frac{\frac{13140911}{125000}}{6 + 2}

Apply the fraction rule:

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

= \frac{13140911}{125000 ( 6 + 2 )}

Rationalize

\frac{13140911}{125000 ( 6 + 2 )} : \frac{13140911 ( 6 - 2 )}{500000}

\frac{13140911}{125000 ( 6 + 2 )}

Multiply by the conjugate

\frac{6 - 2}{6 - 2}

= \frac{13140911 ( 6 - 2 )}{125000 ( 6 + 2 ) ( 6 - 2 )}

125000 (6 + 2) (6 - 2) = 500000

125000 (6 + 2) (6 - 2)

Expand

(6 + 2) (6 - 2) : 4

(6 + 2) (6 - 2)

Apply Difference of Two Squares Formula:

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

a = 6, b = 2

= (6)^{2} - (2)^{2}

Simplify

(6)^{2} - (2)^{2} : 4

(6)^{2} - (2)^{2}

(6)^{2} = 6

(6)^{2}

Apply radical rule:

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

= (6^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 6^{\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

= 6

(2)^{2} = 2

(2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\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

= 2

= 6 - 2

Subtract the numbers:

6 - 2 = 4

= 4

= 4

= 125000 \cdot 4

Expand

125000 \cdot 4 : 500000

125000 \cdot 4

Distribute parentheses

= 125000 \cdot 4

Multiply the numbers:

125000 \cdot 4 = 500000

= 500000

= 500000

= \frac{13140911 ( 6 - 2 )}{500000}

= \frac{13140911 ( 6 - 2 )}{500000}

= \frac{13140911 ( 6 - 2 )}{500000}

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Frequently Asked Questions (FAQ)

What is the value of ((2.68*9.80665))/(sin(75)) ?
The value of ((2.68*9.80665))/(sin(75)) is (13140911(sqrt(6)-sqrt(2)))/(500000)