Solve (1/4)*cos16-2cos5+(1/3)cos27

$$( 1 / 4 ) \cdot \cos 16 - 2 \cos 5 + ( 1 / 3 ) \cos 27$$

Evaluate

$$-\frac{174608615696334637}{120000000000000000}\approx -1.455071797$$

Solution Steps

Evaluate trigonometric functions in the problem

$${(\frac{1}{4})} \cdot 0.9612616959383189 - 2 \cdot 0.9961946980917455 + {(\frac{1}{3})} 0.8910065241883679$$

Convert decimal number 0.9612616959383189 to fraction \frac{9612616959383189}{10000000000}. Reduce the fraction \frac{9612616959383189}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{1}{4}\times \frac{9612616959383189}{10000000000000000}-2\times 0.9961946980917455+\frac{1}{3}\times 0.8910065241883679$$

Multiply \frac{1}{4} times \frac{9612616959383189}{10000000000000000} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 9612616959383189}{4\times 10000000000000000}-2\times 0.9961946980917455+\frac{1}{3}\times 0.8910065241883679$$

Do the multiplications in the fraction \frac{1\times 9612616959383189}{4\times 10000000000000000}.

$$\frac{9612616959383189}{40000000000000000}-2\times 0.9961946980917455+\frac{1}{3}\times 0.8910065241883679$$

Multiply 2 and 0.9961946980917455 to get 1.992389396183491.

$$\frac{9612616959383189}{40000000000000000}-1.992389396183491+\frac{1}{3}\times 0.8910065241883679$$

Convert decimal number 1.992389396183491 to fraction \frac{1992389396183491}{10000000000}. Reduce the fraction \frac{1992389396183491}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{9612616959383189}{40000000000000000}-\frac{1992389396183491}{1000000000000000}+\frac{1}{3}\times 0.8910065241883679$$

Least common multiple of 40000000000000000 and 1000000000000000 is 40000000000000000. Convert \frac{9612616959383189}{40000000000000000} and \frac{1992389396183491}{1000000000000000} to fractions with denominator 40000000000000000.

$$\frac{9612616959383189}{40000000000000000}-\frac{79695575847339640}{40000000000000000}+\frac{1}{3}\times 0.8910065241883679$$

Since \frac{9612616959383189}{40000000000000000} and \frac{79695575847339640}{40000000000000000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9612616959383189-79695575847339640}{40000000000000000}+\frac{1}{3}\times 0.8910065241883679$$

Subtract 79695575847339640 from 9612616959383189 to get -70082958887956451.

$$-\frac{70082958887956451}{40000000000000000}+\frac{1}{3}\times 0.8910065241883679$$

Convert decimal number 0.8910065241883679 to fraction \frac{8910065241883679}{10000000000}. Reduce the fraction \frac{8910065241883679}{10000000000} to lowest terms by extracting and canceling out 1.

$$-\frac{70082958887956451}{40000000000000000}+\frac{1}{3}\times \frac{8910065241883679}{10000000000000000}$$

Multiply \frac{1}{3} times \frac{8910065241883679}{10000000000000000} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{70082958887956451}{40000000000000000}+\frac{1\times 8910065241883679}{3\times 10000000000000000}$$

Do the multiplications in the fraction \frac{1\times 8910065241883679}{3\times 10000000000000000}.

$$-\frac{70082958887956451}{40000000000000000}+\frac{8910065241883679}{30000000000000000}$$

Least common multiple of 40000000000000000 and 30000000000000000 is 120000000000000000. Convert -\frac{70082958887956451}{40000000000000000} and \frac{8910065241883679}{30000000000000000} to fractions with denominator 120000000000000000.

$$-\frac{210248876663869353}{120000000000000000}+\frac{35640260967534716}{120000000000000000}$$

Since -\frac{210248876663869353}{120000000000000000} and \frac{35640260967534716}{120000000000000000} have the same denominator, add them by adding their numerators.

$$\frac{-210248876663869353+35640260967534716}{120000000000000000}$$

Add -210248876663869353 and 35640260967534716 to get -174608615696334637.

$$-\frac{174608615696334637}{120000000000000000}$$

Factor

$$-\frac{174608615696334637}{120000000000000000} = -1\frac{54608615696334620}{1.2 \times 10^{17}} = -1.4550717974694551$$