What is the value of cos((19pi)/(30))cos(pi/5)-sin((19pi)/(30))sin(pi/5) ?

The value of cos((19pi)/(30))cos(pi/5)-sin((19pi)/(30))sin(pi/5) is -(sqrt(3))/2

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cos((19pi)/(30))cos(pi/5)-sin((19pi)/(30))sin(pi/5)

Solution

cos (\frac{19 π}{30}) cos (\frac{π}{5}) - sin (\frac{19 π}{30}) sin (\frac{π}{5})

Solution

- \frac{3}{2}

Decimal Notation

- 0.86602 \dots

Solution steps

cos (\frac{19 π}{30}) cos (\frac{π}{5}) - sin (\frac{19 π}{30}) sin (\frac{π}{5})

Rewrite using trig identities:

cos (\frac{5 π}{6})

cos (\frac{19 π}{30}) cos (\frac{π}{5}) - sin (\frac{19 π}{30}) sin (\frac{π}{5})

Use the Angle Sum identity:

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

= cos (\frac{19 π}{30} + \frac{π}{5})

Simplify:

\frac{19 π}{30} + \frac{π}{5} = \frac{5 π}{6}

\frac{19 π}{30} + \frac{π}{5}

Least Common Multiplier of

30, 5 : 30

30, 5

Least Common Multiplier (LCM)

Prime factorization of

30 : 2 \cdot 3 \cdot 5

30

30

divides by

230 = 15 \cdot 2

= 2 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 5

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

Multiply each factor the greatest number of times it occurs in either

30

5

= 2 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 3 \cdot 5 = 30

= 30

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

30

For

\frac{π}{5} :

multiply the denominator and numerator by

6

\frac{π}{5} = \frac{π 6}{5 \cdot 6} = \frac{π 6}{30}

= \frac{19 π}{30} + \frac{π 6}{30}

Since the denominators are equal, combine the fractions:

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

= \frac{19 π + π 6}{30}

Add similar elements:

19 π + 6 π = 25 π

= \frac{25 π}{30}

Cancel the common factor:

5

= \frac{5 π}{6}

= cos (\frac{5 π}{6})

= cos (\frac{5 π}{6})

Use the following trivial identity:

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 π}{6})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{3}{2}

= - \frac{3}{2}

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Frequently Asked Questions (FAQ)

What is the value of cos((19pi)/(30))cos(pi/5)-sin((19pi)/(30))sin(pi/5) ?
The value of cos((19pi)/(30))cos(pi/5)-sin((19pi)/(30))sin(pi/5) is -(sqrt(3))/2