What is the value of sin(pi/5)cos(pi/(20))+cos(pi/5)sin(pi/(20)) ?

The value of sin(pi/5)cos(pi/(20))+cos(pi/5)sin(pi/(20)) is (sqrt(2))/2

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sin(pi/5)cos(pi/(20))+cos(pi/5)sin(pi/(20))

Solution

sin (\frac{π}{5}) cos (\frac{π}{20}) + cos (\frac{π}{5}) sin (\frac{π}{20})

Solution

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

sin (\frac{π}{5}) cos (\frac{π}{20}) + cos (\frac{π}{5}) sin (\frac{π}{20})

Rewrite using trig identities:

sin (\frac{π}{4})

sin (\frac{π}{5}) cos (\frac{π}{20}) + cos (\frac{π}{5}) sin (\frac{π}{20})

Use the Angle Sum identity:

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

= sin (\frac{π}{5} + \frac{π}{20})

Simplify:

\frac{π}{5} + \frac{π}{20} = \frac{π}{4}

\frac{π}{5} + \frac{π}{20}

Least Common Multiplier of

5, 20 : 20

5, 20

Least Common Multiplier (LCM)

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

Prime factorization of

20 : 2 \cdot 2 \cdot 5

20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5

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

5

20

= 5 \cdot 2 \cdot 2

Multiply the numbers:

5 \cdot 2 \cdot 2 = 20

= 20

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

20

For

\frac{π}{5} :

multiply the denominator and numerator by

4

\frac{π}{5} = \frac{π 4}{5 \cdot 4} = \frac{π 4}{20}

= \frac{π 4}{20} + \frac{π}{20}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + π}{20}

Add similar elements:

4 π + π = 5 π

= \frac{5 π}{20}

Cancel the common factor:

5

= \frac{π}{4}

= sin (\frac{π}{4})

= sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2}

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Frequently Asked Questions (FAQ)

What is the value of sin(pi/5)cos(pi/(20))+cos(pi/5)sin(pi/(20)) ?
The value of sin(pi/5)cos(pi/(20))+cos(pi/5)sin(pi/(20)) is (sqrt(2))/2