Solve frac{sin50^{circ}}{sin40^{circ}}+csc40^circ/sec50^circ-4cos50^circcsc40^circ

$$\frac { \sin 50 ^ { \circ } } { \sin 40 ^ { \circ } } + \frac { \csc 40 ^ { \circ } } { \sec 50 ^ { \circ } } - 4 \cos 50 ^ { \circ } \csc 40 ^ { \circ }$$

Evaluate

$$-\frac{1130154004628619334576967459059172849408252457683727455175040829}{624999999999999966348287098845868750000000000000000000000000000}\approx -1.808246407$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{0.766044443118978}{0.6427876096865393} + \frac{1.5557238268604126}{1.5557238268604123} - 4 \cdot 0.6427876096865394 \cdot 1.5557238268604126$$

Expand \frac{0.766044443118978}{0.6427876096865393} by multiplying both numerator and the denominator by 10000000000000000.

$$\frac{7660444431189780}{6427876096865393}+\frac{1.5557238268604126}{1.5557238268604123}-4\times 0.6427876096865394\times 1.5557238268604126$$

Expand \frac{1.5557238268604126}{1.5557238268604123} by multiplying both numerator and the denominator by 10000000000000000.

$$\frac{7660444431189780}{6427876096865393}+\frac{15557238268604126}{15557238268604123}-4\times 0.6427876096865394\times 1.5557238268604126$$

Least common multiple of 6427876096865393 and 15557238268604123 is 99999999999999994615725935815339. Convert \frac{7660444431189780}{6427876096865393} and \frac{15557238268604126}{15557238268604123} to fractions with denominator 99999999999999994615725935815339.

$$\frac{119175359259420988857604703462940}{99999999999999994615725935815339}+\frac{100000000000000013899354226411518}{99999999999999994615725935815339}-4\times 0.6427876096865394\times 1.5557238268604126$$

Since \frac{119175359259420988857604703462940}{99999999999999994615725935815339} and \frac{100000000000000013899354226411518}{99999999999999994615725935815339} have the same denominator, add them by adding their numerators.

$$\frac{119175359259420988857604703462940+100000000000000013899354226411518}{99999999999999994615725935815339}-4\times 0.6427876096865394\times 1.5557238268604126$$

Add 119175359259420988857604703462940 and 100000000000000013899354226411518 to get 219175359259421002756958929874458.

$$\frac{219175359259421002756958929874458}{99999999999999994615725935815339}-4\times 0.6427876096865394\times 1.5557238268604126$$

Multiply 4 and 0.6427876096865394 to get 2.5711504387461576.

$$\frac{219175359259421002756958929874458}{99999999999999994615725935815339}-2.5711504387461576\times 1.5557238268604126$$

Multiply 2.5711504387461576 and 1.5557238268604126 to get 4.00000000000000117826369980062576.

$$\frac{219175359259421002756958929874458}{99999999999999994615725935815339}-4.00000000000000117826369980062576$$

Convert decimal number 4.00000000000000117826369980062576 to fraction \frac{25000000000000007364148123753911}{10000000000}. Reduce the fraction \frac{25000000000000007364148123753911}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{219175359259421002756958929874458}{99999999999999994615725935815339}-\frac{25000000000000007364148123753911}{6250000000000000000000000000000}$$

Least common multiple of 99999999999999994615725935815339 and 6250000000000000000000000000000 is 624999999999999966348287098845868750000000000000000000000000000. Convert \frac{219175359259421002756958929874458}{99999999999999994615725935815339} and \frac{25000000000000007364148123753911}{6250000000000000000000000000000} to fractions with denominator 624999999999999966348287098845868750000000000000000000000000000.

$$\frac{1369845995371381267230993311715362500000000000000000000000000000}{624999999999999966348287098845868750000000000000000000000000000}-\frac{2500000000000000601807960770774535349408252457683727455175040829}{624999999999999966348287098845868750000000000000000000000000000}$$

Since \frac{1369845995371381267230993311715362500000000000000000000000000000}{624999999999999966348287098845868750000000000000000000000000000} and \frac{2500000000000000601807960770774535349408252457683727455175040829}{624999999999999966348287098845868750000000000000000000000000000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1369845995371381267230993311715362500000000000000000000000000000-2500000000000000601807960770774535349408252457683727455175040829}{624999999999999966348287098845868750000000000000000000000000000}$$

Subtract 2500000000000000601807960770774535349408252457683727455175040829 from 1369845995371381267230993311715362500000000000000000000000000000 to get -1130154004628619334576967459059172849408252457683727455175040829.

$$-\frac{1130154004628619334576967459059172849408252457683727455175040829}{624999999999999966348287098845868750000000000000000000000000000}$$