cos 20°+cos 40°+cos60°+....
cos180°=
Answers
Given to find the value of :-
cos20° + cos40° + cos60° +... cos180°
To know :-
cos(180-θ) = -cosθ
cos180° = cos(90°+90°)
= -sin 90 [cos(90+θ) = sinθ]
cos 180°= -1
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Solution:-
cos20° + cos40° + cos60° + cos80° + cos100° + cos120° + cos 140° + cos160° + cos180°
cos20° can be written as cos(180°-160°)
cos40° can be written as cos(180°-140°)
cos60° can be written as cos(180°-120°)
cos80° can be written as cos(180°-100°)
If you observe these I have written all interms of after terms that means 160°, 140°, 120°, 100°
So,
cos20° + cos40° + cos60° + cos80° + cos100° + cos120° + cos 140° + cos160° + cos180°
cos(180°-160°) + cos(180°-140°) + cos(180°-120°) + cos(180°-100°) + cos100° + cos120° + cos 140° + cos160° + cos180°
As we know ,
cos(180-θ) = -cosθ
= -cos160° - cos140° -cos120° - cos100° + cos100° + cos120° + cos140° + cos160° + cos180°
= -cos160° + cos160° -cos140° +cos140° -cos120° +cos120° -cos100°+cos100° + cos180°
All cancelled except cos180°
= cos180°
= -1
Note :-
The purpose of writing 20°,40° 60° in terms of 120°,140° by using Quadrant angles is to be cancelled opposite signs And we can get the value