Solve frac{1+cos(28^{circ})}{2sin(20^circ)}-sin(10^circ)(1/tan5^circ-tan5^circ)

$$\frac { 1 + \cos ( 28 ^ { \circ } ) } { 2 \sin ( 20 ^ { \circ } ) } - \sin ( 10 ^ { \circ } ) ( \frac { 1 } { \tan 5 ^ { \circ } } - \tan 5 ^ { \circ } )$$

Evaluate

$$\frac{33473864919339540926052026870019775554053923353028753991981760353}{42746978912153904889369312336410000000000000000000000000000000000}\approx 0.783069723$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{1 + 0.882947592858927}{2 \cdot 0.3420201433256687} - 0.17364817766693033 {(\frac{1}{0.08748866352592401} - 0.08748866352592401)}$$

Add 1 and 0.882947592858927 to get 1.882947592858927.

$$\frac{1.882947592858927}{2\times 0.3420201433256687}-0.17364817766693033\left(\frac{1}{0.08748866352592401}-0.08748866352592401\right)$$

Multiply 2 and 0.3420201433256687 to get 0.6840402866513374.

$$\frac{1.882947592858927}{0.6840402866513374}-0.17364817766693033\left(\frac{1}{0.08748866352592401}-0.08748866352592401\right)$$

Expand \frac{1.882947592858927}{0.6840402866513374} by multiplying both numerator and the denominator by 10000000000000000.

$$\frac{18829475928589270}{6840402866513374}-0.17364817766693033\left(\frac{1}{0.08748866352592401}-0.08748866352592401\right)$$

Reduce the fraction \frac{18829475928589270}{6840402866513374} to lowest terms by extracting and canceling out 2.

$$\frac{9414737964294635}{3420201433256687}-0.17364817766693033\left(\frac{1}{0.08748866352592401}-0.08748866352592401\right)$$

Expand \frac{1}{0.08748866352592401} by multiplying both numerator and the denominator by 100000000000000000.

$$\frac{9414737964294635}{3420201433256687}-0.17364817766693033\left(\frac{100000000000000000}{8748866352592401}-0.08748866352592401\right)$$

Convert decimal number 0.08748866352592401 to fraction \frac{8748866352592401}{10000000000}. Reduce the fraction \frac{8748866352592401}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{9414737964294635}{3420201433256687}-0.17364817766693033\left(\frac{100000000000000000}{8748866352592401}-\frac{8748866352592401}{100000000000000000}\right)$$

Least common multiple of 8748866352592401 and 100000000000000000 is 874886635259240100000000000000000. Convert \frac{100000000000000000}{8748866352592401} and \frac{8748866352592401}{100000000000000000} to fractions with denominator 874886635259240100000000000000000.

$$\frac{9414737964294635}{3420201433256687}-0.17364817766693033\left(\frac{10000000000000000000000000000000000}{874886635259240100000000000000000}-\frac{76542662455523462255933242944801}{874886635259240100000000000000000}\right)$$

Since \frac{10000000000000000000000000000000000}{874886635259240100000000000000000} and \frac{76542662455523462255933242944801}{874886635259240100000000000000000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9414737964294635}{3420201433256687}-0.17364817766693033\times \frac{10000000000000000000000000000000000-76542662455523462255933242944801}{874886635259240100000000000000000}$$

Subtract 76542662455523462255933242944801 from 10000000000000000000000000000000000 to get 9923457337544476537744066757055199.

$$\frac{9414737964294635}{3420201433256687}-0.17364817766693033\times \frac{9923457337544476537744066757055199}{874886635259240100000000000000000}$$

Convert decimal number 0.17364817766693033 to fraction \frac{17364817766693033}{10000000000}. Reduce the fraction \frac{17364817766693033}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{9414737964294635}{3420201433256687}-\frac{17364817766693033}{100000000000000000}\times \frac{9923457337544476537744066757055199}{874886635259240100000000000000000}$$

Multiply \frac{17364817766693033}{100000000000000000} times \frac{9923457337544476537744066757055199}{874886635259240100000000000000000} by multiplying numerator times numerator and denominator times denominator.

$$\frac{9414737964294635}{3420201433256687}-\frac{17364817766693033\times 9923457337544476537744066757055199}{100000000000000000\times 874886635259240100000000000000000}$$

Do the multiplications in the fraction \frac{17364817766693033\times 9923457337544476537744066757055199}{100000000000000000\times 874886635259240100000000000000000}.

$$\frac{9414737964294635}{3420201433256687}-\frac{172319028282012668406798801192384509254707669728567}{87488663525924010000000000000000000000000000000000}$$

Reduce the fraction \frac{172319028282012668406798801192384509254707669728567}{87488663525924010000000000000000000000000000000000} to lowest terms by extracting and canceling out 7.

$$\frac{9414737964294635}{3420201433256687}-\frac{24617004040287524058114114456054929893529667104081}{12498380503703430000000000000000000000000000000000}$$

Least common multiple of 3420201433256687 and 12498380503703430000000000000000000000000000000000 is 42746978912153904889369312336410000000000000000000000000000000000. Convert \frac{9414737964294635}{3420201433256687} and \frac{24617004040287524058114114456054929893529667104081}{12498380503703430000000000000000000000000000000000} to fractions with denominator 42746978912153904889369312336410000000000000000000000000000000000.

$$\frac{117668977420416585357725180098050000000000000000000000000000000000}{42746978912153904889369312336410000000000000000000000000000000000}-\frac{84195112501077044431673153228030224445946076646971246008018239647}{42746978912153904889369312336410000000000000000000000000000000000}$$

Since \frac{117668977420416585357725180098050000000000000000000000000000000000}{42746978912153904889369312336410000000000000000000000000000000000} and \frac{84195112501077044431673153228030224445946076646971246008018239647}{42746978912153904889369312336410000000000000000000000000000000000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{117668977420416585357725180098050000000000000000000000000000000000-84195112501077044431673153228030224445946076646971246008018239647}{42746978912153904889369312336410000000000000000000000000000000000}$$

Subtract 84195112501077044431673153228030224445946076646971246008018239647 from 117668977420416585357725180098050000000000000000000000000000000000 to get 33473864919339540926052026870019775554053923353028753991981760353.

$$\frac{33473864919339540926052026870019775554053923353028753991981760353}{42746978912153904889369312336410000000000000000000000000000000000}$$

Factor

$$\frac{33473864919339540926052026870019775554053923353028753991981760353}{3 \cdot 7 \cdot 179 \cdot 2999 \cdot 25693 \cdot 40151 \cdot 494267 \cdot 743676121 \cdot 2 ^ {34} \cdot 5 ^ {34}} = 0.7830697226142964$$