There is a special solution to this problem.

Use vectors to combine numbers and shapes

Cosθ can be understood as the projection of unit vector E (the angle between E and X axis is θ) on X axis, that is, ex.

It is observed that the numbers 23, 95, 167, 239, 3 1 1 are arithmetic progression and can be abstracted as θ, θ+72, θ+72×2, θ+72×3, θ+72×3, θ+72.

Let the unit vectors e 1, e2, e3, e4 and e5 form angles with the x axis of 23, 95, 167, 239, 3 1 1 respectively.

You can draw a picture. After translation, we can know that the vectors are end to end, so e 1+e2+e3+e4+e5=0 (zero vector).

e 1+e2+e3+e4+e5？ Projection on the x axis =0

(e 1+e2+e3+e4+e5？ ) is projected on the x axis.

= e65438+0 projection on the x axis+e2 projection on the x axis+... +e5 projection on the x axis.

= cos 23+cos 95+cos 167+cos 239+cos 3 1 1

=0?

You can also draw general conclusions.

cosθ+cos(θ+72)+cos(θ+72×2)+cos(θ+72×3)+cos(θ+72×4)= 0

cosθ+cos(θ+ 120)+cos(θ- 120)= 0

etc