What is the value of cos((2pi)/(15))cos(pi/(30))-sin((2pi)/(15))sin(pi/(30)) ?

The value of cos((2pi)/(15))cos(pi/(30))-sin((2pi)/(15))sin(pi/(30)) is (sqrt(3))/2

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Popular >

cos((2pi)/(15))cos(pi/(30))-sin((2pi)/(15))sin(pi/(30))

Solution

cos (\frac{2 π}{15}) cos (\frac{π}{30}) - sin (\frac{2 π}{15}) sin (\frac{π}{30})

Solution

\frac{3}{2}

Decimal Notation

0.86602 \dots

Solution steps

cos (\frac{2 π}{15}) cos (\frac{π}{30}) - sin (\frac{2 π}{15}) sin (\frac{π}{30})

Rewrite using trig identities:

cos (\frac{π}{6})

cos (\frac{2 π}{15}) cos (\frac{π}{30}) - sin (\frac{2 π}{15}) sin (\frac{π}{30})

Use the Angle Sum identity:

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

= cos (\frac{2 π}{15} + \frac{π}{30})

Simplify:

\frac{2 π}{15} + \frac{π}{30} = \frac{π}{6}

\frac{2 π}{15} + \frac{π}{30}

Least Common Multiplier of

15, 30 : 30

15, 30

Least Common Multiplier (LCM)

Prime factorization of

15 : 3 \cdot 5

15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 5

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

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

15

30

= 3 \cdot 5 \cdot 2

Multiply the numbers:

3 \cdot 5 \cdot 2 = 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{2 π}{15} :

multiply the denominator and numerator by

2

\frac{2 π}{15} = \frac{2 π 2}{15 \cdot 2} = \frac{4 π}{30}

= \frac{4 π}{30} + \frac{π}{30}

Since the denominators are equal, combine the fractions:

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

= \frac{4 π + π}{30}

Add similar elements:

4 π + π = 5 π

= \frac{5 π}{30}

Cancel the common factor:

5

= \frac{π}{6}

= cos (\frac{π}{6})

= cos (\frac{π}{6})

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{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}

Popular Examples

(28)/(tan(\frac{7pi){18})}

\frac{28}{tan ( \frac{7 π}{18} )}

(120sin(8))/(30)

\frac{120 sin ( 8 )}{30}

arcsin(((2*5.8*10^{-7}))/(5.3*10^{-4)})

arcsin (\frac{( 2 \cdot 5.8 \cdot 1 0 ^{- 7} )}{5.3 \cdot 1 0 ^{- 4}})

cos(65.7)

cos (65. 7^{\circ})

120(tan(32.3))

120 (tan (32. 3^{\circ}))

Frequently Asked Questions (FAQ)

What is the value of cos((2pi)/(15))cos(pi/(30))-sin((2pi)/(15))sin(pi/(30)) ?
The value of cos((2pi)/(15))cos(pi/(30))-sin((2pi)/(15))sin(pi/(30)) is (sqrt(3))/2