Question

Cos20°+ cos100° + cos140° = ?

Answer 1

Answer:

To prove:  cos(20�) + cos(100�) +cos(140�) = 0

Write the angles in the first two terms in terms  

of their average, which is 60�, a special angle,

and use a double angle identity on each:

cos(20�) = cos(60�-40�) = cos(60�)cos(40�)+sin(60�)sin(40�)

cos(100�) = cos(60�+40�) = cos(60�)cos(40�)-sin(60�)sin(40�)

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 SUM OF FIRST TWO TERMS = 2cos(60�)cos(40�) = 2%281%2F2%29cos(40�) = cos(40�)  

Write the angle in the third term in terms of 40� and use

use a double angle identity:  

cos(140�) = cos(180�-40�) = cos(180�)cos(40�)+sin(180�)sin(40�)

                         = (-1)cos(40�) + 0(sin(40�)

                         = -cos(40�)

So we have:

cos(20�) + cos(100�) + cos(140�) = cos(40�) - cos(40�) = 0