What is the value of 32.09(1+0.0053sin^2(45)) ?

The value of 32.09(1+0.0053sin^2(45)) is 64350077/2000000

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32.09(1+0.0053sin^2(45))

Solution

32.09 (1 + 0.0053 sin^{2} (4 5^{\circ}))

Solution

\frac{64350077}{2000000}

Decimal Notation

32.1750385

Solution steps

32.09 (1 + 0.0053 sin^{2} (4 5^{\circ}))

= \frac{3209}{100} (1 + \frac{53}{10000} sin^{2} (4 5^{\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}

= \frac{3209}{100} 1 + \frac{53}{10000} (\frac{2}{2})^{2}

Simplify

\frac{3209}{100} 1 + \frac{53}{10000} (\frac{2}{2})^{2} : \frac{64350077}{2000000}

\frac{3209}{100} 1 + \frac{53}{10000} (\frac{2}{2})^{2}

\frac{53}{10000} (\frac{2}{2})^{2} = \frac{53}{20000}

\frac{53}{10000} (\frac{2}{2})^{2}

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

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

Apply exponent rule:

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

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

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

Cancel the common factor:

2

= \frac{1}{2}

= \frac{1}{2} \cdot \frac{53}{10000}

Multiply fractions:

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

= \frac{53 \cdot 1}{10000 \cdot 2}

Multiply the numbers:

53 \cdot 1 = 53

= \frac{53}{10000 \cdot 2}

Multiply the numbers:

10000 \cdot 2 = 20000

= \frac{53}{20000}

= \frac{3209}{100} (\frac{53}{20000} + 1)

Join

1 + \frac{53}{20000} : \frac{20053}{20000}

1 + \frac{53}{20000}

Convert element to fraction:

1 = \frac{1 \cdot 20000}{20000}

= \frac{1 \cdot 20000}{20000} + \frac{53}{20000}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 20000 + 53}{20000}

1 \cdot 20000 + 53 = 20053

1 \cdot 20000 + 53

Multiply the numbers:

1 \cdot 20000 = 20000

= 20000 + 53

Add the numbers:

20000 + 53 = 20053

= 20053

= \frac{20053}{20000}

= \frac{3209}{100} \cdot \frac{20053}{20000}

Multiply fractions:

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

= \frac{3209 \cdot 20053}{100 \cdot 20000}

Multiply the numbers:

3209 \cdot 20053 = 64350077

= \frac{64350077}{100 \cdot 20000}

Multiply the numbers:

100 \cdot 20000 = 2000000

= \frac{64350077}{2000000}

= \frac{64350077}{2000000}

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Frequently Asked Questions (FAQ)

What is the value of 32.09(1+0.0053sin^2(45)) ?
The value of 32.09(1+0.0053sin^2(45)) is 64350077/2000000