# Which is the value of the sum cos 75 + cos 15?

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

cos75 + cos15 = ?

We know that:

cosx + cosy = 2cos(x+y)/2 * cos(x-y)/2

Then :

cos75+cos15= 2 cos(75+15)/2 * cos(75-15)/2

= 2cos(90/2) * cos(60/2)

= 2cos(45)*cos(30)

= 2(sqrt2)/2) * sqrt(3)/2

= sqrt(6) /2

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll consider the fact that being an addition of two alike functions, we'll transform the addition into a product, in this way:

cos a + cos b = 2cos [(a+b)/2]cos [(a-b)/2]

cos 75 + cos 15 = 2cos[(75+15)/2] cos [(75-15)/2]

cos 75 + cos 15 = 2cos45cos30

cos 75 + cos 15 = 2*[(sqrt2)/2]*[(sqrt3)/2]

cos 75 + cos 15 = sqrt(2*3)/2=sqrt(6)/2

Another manner of solving would be to write the angles:

15 = 45 - 30

75 = 45 + 30

cos (45 - 30) = cos45*cos30 + sin30*sin45

=(sqrt2/2)(sqrt3/2) + sqrt2/4

= (sqrt6 + sqrt2)/4

cos (45 + 30) = cos45*cos30 - sin30*sin45

= (sqrt6 - sqrt2)/4

So,

cos 75 + cos 15 = (sqrt6 - sqrt2+sqrt6 + sqrt2)/4

sin 15 + sin 75 = 2sqrt6/4

sin 15 + sin 75 = sqrt6/2

neela | High School Teacher | (Level 3) Valedictorian

Posted on

To determine cos75+cos15.

Solution:

cos75+cos15 = 2(cos(75+15)/2}{cos(75-15)/2}, as cosA+cosB = 2cos(A+B)/2 cos(A-B)/2.

= 2 cos45*cos30

=2(1/sqrt2)(sqrt3/2)

=sqrt(3/2) = (sqrt6)/2

=