What is the value of 12.4-10(0.5pi-arcsin(0.5)-0.5(1-(0.5)^2)^{1/2}) ?

The value of 12.4-10(0.5pi-arcsin(0.5)-0.5(1-(0.5)^2)^{1/2}) is 62/5-(5(4pi-3*3^{1/2}))/6

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12.4-10(0.5pi-arcsin(0.5)-0.5(1-(0.5)^2)^{1/2})

Solution

12.4 - 10 (0.5 π - arcsin (0.5) - 0.5 (1 - (0.5)^{2})^{\frac{1}{2}})

Solution

\frac{62}{5} - \frac{5 ( 4 π - 3 \cdot 3 ^{\frac{1}{2}} )}{6}

Decimal Notation

358.57

Solution steps

12.4 - 10 (0.5 π - arcsin (0.5) - 0.5 (1 - (0.5)^{2})^{\frac{1}{2}})

= \frac{62}{5} - 10 \frac{1}{2} π - arcsin (\frac{1}{2}) - \frac{1}{2} (1 - (\frac{1}{2})^{2})^{\frac{1}{2}}

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{6}

= \frac{62}{5} - 10 \frac{1}{2} π - \frac{π}{6} - \frac{1}{2} (1 - (\frac{1}{2})^{2})^{\frac{1}{2}}

Simplify

\frac{62}{5} - 10 \frac{1}{2} π - \frac{π}{6} - \frac{1}{2} (1 - (\frac{1}{2})^{2})^{\frac{1}{2}} : \frac{62}{5} - \frac{5 ( 4 π - 3 \cdot 3 ^{\frac{1}{2}} )}{6}

\frac{62}{5} - 10 \frac{1}{2} π - \frac{π}{6} - \frac{1}{2} (1 - (\frac{1}{2})^{2})^{\frac{1}{2}}

10 \frac{1}{2} π - \frac{π}{6} - \frac{1}{2} (1 - (\frac{1}{2})^{2})^{\frac{1}{2}} = \frac{5 ( 4 π - 3 \cdot 3 ^{\frac{1}{2}} )}{6}

10 \frac{1}{2} π - \frac{π}{6} - \frac{1}{2} (1 - (\frac{1}{2})^{2})^{\frac{1}{2}}

\frac{1}{2} π = \frac{π}{2}

\frac{1}{2} π

Multiply fractions:

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

= \frac{1 π}{2}

Multiply:

1 π = π

= \frac{π}{2}

\frac{1}{2} (1 - (\frac{1}{2})^{2})^{\frac{1}{2}} = \frac{3 ^{\frac{1}{2}}}{4}

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

(1 - (\frac{1}{2})^{2})^{\frac{1}{2}} = \frac{3 ^{\frac{1}{2}}}{2}

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

(\frac{1}{2})^{2} = \frac{1}{4}

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= (1 - \frac{1}{4})^{\frac{1}{2}}

Join

1 - \frac{1}{4} : \frac{3}{4}

1 - \frac{1}{4}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 1

Subtract the numbers:

4 - 1 = 3

= 3

= \frac{3}{4}

= (\frac{3}{4})^{\frac{1}{2}}

Apply radical rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}},

assuming

a \geq 0, b \geq 0

= \frac{3 ^{\frac{1}{2}}}{4 ^{\frac{1}{2}}}

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

4^{\frac{1}{2}}

Factor the number:

4 = 2^{2}

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

Apply exponent rule:

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

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

= 2

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

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

Multiply fractions:

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

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

Multiply:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 ^{\frac{1}{2}}}{4}

= 10 (\frac{π}{2} - \frac{π}{6} - \frac{3 ^{\frac{1}{2}}}{4})

Join

\frac{π}{2} - \frac{π}{6} - \frac{3 ^{\frac{1}{2}}}{4} : \frac{4 π - 3 \cdot 3 ^{\frac{1}{2}}}{12}

\frac{π}{2} - \frac{π}{6} - \frac{3 ^{\frac{1}{2}}}{4}

Least Common Multiplier of

2, 6, 4 : 12

2, 6, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

2, 6, 4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{π}{2} :

multiply the denominator and numerator by

6

\frac{π}{2} = \frac{π 6}{2 \cdot 6} = \frac{π 6}{12}

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{3 ^{\frac{1}{2}}}{4} :

multiply the denominator and numerator by

3

\frac{3 ^{\frac{1}{2}}}{4} = \frac{3 ^{\frac{1}{2}} \cdot 3}{4 \cdot 3} = \frac{3 ^{\frac{1}{2}} \cdot 3}{12}

= \frac{π 6}{12} - \frac{π 2}{12} - \frac{3 ^{\frac{1}{2}} \cdot 3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 6 - π 2 - 3 ^{\frac{1}{2}} \cdot 3}{12}

Add similar elements:

6 π - 2 π = 4 π

= \frac{4 π - 3 \cdot 3 ^{\frac{1}{2}}}{12}

= 10 \cdot \frac{4 π - 3 \cdot 3 ^{\frac{1}{2}}}{12}

Multiply fractions:

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

= \frac{( 4 π - 3 ^{\frac{1}{2}} \cdot 3 ) \cdot 10}{12}

Cancel the common factor:

2

= \frac{5 ( 4 π - 3 \cdot 3 ^{\frac{1}{2}} )}{6}

= \frac{62}{5} - \frac{5 ( 4 π - 3 \cdot 3 ^{\frac{1}{2}} )}{6}

= \frac{62}{5} - \frac{5 ( 4 π - 3 \cdot 3 ^{\frac{1}{2}} )}{6}

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Frequently Asked Questions (FAQ)

What is the value of 12.4-10(0.5pi-arcsin(0.5)-0.5(1-(0.5)^2)^{1/2}) ?
The value of 12.4-10(0.5pi-arcsin(0.5)-0.5(1-(0.5)^2)^{1/2}) is 62/5-(5(4pi-3*3^{1/2}))/6