What is the value of cos((3pi)/7)cos((2pi}{21})+sin((3pi)/7)sin(\frac{2pi)/(21)) ?

The value of cos((3pi)/7)cos((2pi}{21})+sin((3pi)/7)sin(\frac{2pi)/(21)) is 1/2

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cos((3pi)/7)cos((2pi}{21})+sin((3pi)/7)sin(\frac{2pi)/(21))

Solution

cos (\frac{3 π}{7}) cos (\frac{2 π}{21}) + sin (\frac{3 π}{7}) sin (\frac{2 π}{21})

Solution

\frac{1}{2}

Decimal Notation

0.5

Solution steps

cos (\frac{3 π}{7}) cos (\frac{2 π}{21}) + sin (\frac{3 π}{7}) sin (\frac{2 π}{21})

Rewrite using trig identities:

cos (\frac{π}{3})

cos (\frac{3 π}{7}) cos (\frac{2 π}{21}) + sin (\frac{3 π}{7}) sin (\frac{2 π}{21})

Use the Angle Difference identity:

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

= cos (\frac{3 π}{7} - \frac{2 π}{21})

Simplify:

\frac{3 π}{7} - \frac{2 π}{21} = \frac{π}{3}

\frac{3 π}{7} - \frac{2 π}{21}

Least Common Multiplier of

7, 21 : 21

7, 21

Least Common Multiplier (LCM)

Prime factorization of

7 : 7

7

7

is a prime number, therefore no factorization is possible

= 7

Prime factorization of

21 : 3 \cdot 7

21

21

divides by

321 = 7 \cdot 3

= 3 \cdot 7

3, 7

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 7

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

7

21

= 7 \cdot 3

Multiply the numbers:

7 \cdot 3 = 21

= 21

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

21

For

\frac{3 π}{7} :

multiply the denominator and numerator by

3

\frac{3 π}{7} = \frac{3 π 3}{7 \cdot 3} = \frac{9 π}{21}

= \frac{9 π}{21} - \frac{2 π}{21}

Since the denominators are equal, combine the fractions:

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

= \frac{9 π - 2 π}{21}

Add similar elements:

9 π - 2 π = 7 π

= \frac{7 π}{21}

Cancel the common factor:

7

= \frac{π}{3}

= cos (\frac{π}{3})

= cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

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{1}{2}

= \frac{1}{2}

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Frequently Asked Questions (FAQ)

What is the value of cos((3pi)/7)cos((2pi}{21})+sin((3pi)/7)sin(\frac{2pi)/(21)) ?
The value of cos((3pi)/7)cos((2pi}{21})+sin((3pi)/7)sin(\frac{2pi)/(21)) is 1/2