What is the value of cos((13pi)/(36))cos(pi/9)+sin((13pi)/(36))sin(pi/9) ?

The value of cos((13pi)/(36))cos(pi/9)+sin((13pi)/(36))sin(pi/9) is (sqrt(2))/2

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cos((13pi)/(36))cos(pi/9)+sin((13pi)/(36))sin(pi/9)

Solution

cos (\frac{13 π}{36}) cos (\frac{π}{9}) + sin (\frac{13 π}{36}) sin (\frac{π}{9})

Solution

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

cos (\frac{13 π}{36}) cos (\frac{π}{9}) + sin (\frac{13 π}{36}) sin (\frac{π}{9})

Rewrite using trig identities:

cos (\frac{π}{4})

cos (\frac{13 π}{36}) cos (\frac{π}{9}) + sin (\frac{13 π}{36}) sin (\frac{π}{9})

Use the Angle Difference identity:

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

= cos (\frac{13 π}{36} - \frac{π}{9})

Simplify:

\frac{13 π}{36} - \frac{π}{9} = \frac{π}{4}

\frac{13 π}{36} - \frac{π}{9}

Least Common Multiplier of

36, 9 : 36

36, 9

Least Common Multiplier (LCM)

Prime factorization of

36 : 2 \cdot 2 \cdot 3 \cdot 3

36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

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

36

9

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

For

\frac{π}{9} :

multiply the denominator and numerator by

4

\frac{π}{9} = \frac{π 4}{9 \cdot 4} = \frac{π 4}{36}

= \frac{13 π}{36} - \frac{π 4}{36}

Since the denominators are equal, combine the fractions:

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

= \frac{13 π - π 4}{36}

Add similar elements:

13 π - 4 π = 9 π

= \frac{9 π}{36}

Cancel the common factor:

9

= \frac{π}{4}

= cos (\frac{π}{4})

= cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

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{2}{2}

= \frac{2}{2}

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Frequently Asked Questions (FAQ)

What is the value of cos((13pi)/(36))cos(pi/9)+sin((13pi)/(36))sin(pi/9) ?
The value of cos((13pi)/(36))cos(pi/9)+sin((13pi)/(36))sin(pi/9) is (sqrt(2))/2