What is the value of cos((13pi)/(15))cos(-pi/5)-sin((13pi)/(15))sin(-pi/5) ?

The value of cos((13pi)/(15))cos(-pi/5)-sin((13pi)/(15))sin(-pi/5) is -1/2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos((13pi)/(15))cos(-pi/5)-sin((13pi)/(15))sin(-pi/5)

Solution

cos (\frac{13 π}{15}) cos (- \frac{π}{5}) - sin (\frac{13 π}{15}) sin (- \frac{π}{5})

Solution

- \frac{1}{2}

Decimal Notation

- 0.5

Solution steps

cos (\frac{13 π}{15}) cos (- \frac{π}{5}) - sin (\frac{13 π}{15}) sin (- \frac{π}{5})

Rewrite using trig identities:

cos (\frac{2 π}{3})

cos (\frac{13 π}{15}) cos (- \frac{π}{5}) - sin (\frac{13 π}{15}) sin (- \frac{π}{5})

Use the Angle Sum identity:

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

= cos (\frac{13 π}{15} - \frac{π}{5})

Simplify:

\frac{13 π}{15} - \frac{π}{5} = \frac{2 π}{3}

\frac{13 π}{15} - \frac{π}{5}

Least Common Multiplier of

15, 5 : 15

15, 5

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

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

15

5

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

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

15

For

\frac{π}{5} :

multiply the denominator and numerator by

3

\frac{π}{5} = \frac{π 3}{5 \cdot 3} = \frac{π 3}{15}

= \frac{13 π}{15} - \frac{π 3}{15}

Since the denominators are equal, combine the fractions:

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

= \frac{13 π - π 3}{15}

Add similar elements:

13 π - 3 π = 10 π

= \frac{10 π}{15}

Cancel the common factor:

5

= \frac{2 π}{3}

= cos (\frac{2 π}{3})

= cos (\frac{2 π}{3})

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 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}

Popular Examples

5cos(45)

5 cos (4 5^{\circ})

sin((3pi)/6)

sin (\frac{3 π}{6})

sin^2(1/2)

sin^{2} (\frac{1}{2})

cot((19pi)/6)

cot (\frac{19 π}{6})

tan((-5pi)/(12))

tan (\frac{- 5 π}{12})

Frequently Asked Questions (FAQ)

What is the value of cos((13pi)/(15))cos(-pi/5)-sin((13pi)/(15))sin(-pi/5) ?
The value of cos((13pi)/(15))cos(-pi/5)-sin((13pi)/(15))sin(-pi/5) is -1/2