Show that cot (-390°)-4 cos 480 = 2-sqrt3.?

1 Answer
Mar 27, 2018

See the method below.

Explanation:

cot(-390˚)-4cos(480˚)=2-sqrt3

Recall that cot(x) = 1/tanx = cosx/sinx
Since all trigonometric functions repeat after 360˚ we can add or subtract 360˚ to the angles without changing anything. Doing this we get

cot(-360˚ -30˚)-4cos(360˚ + 120˚)=2-sqrt3=
cot(-30˚)-4cos(120˚)=2-sqrt3

We can write cotx as cosx/sinx

cos(-30˚)/sin(-30˚)-4cos(120˚)=2-sqrt3

This can be further simplified as cos(-x)= cos(x) and sin(-x)=-sin(x).

-cos(30˚)/sin(30˚)-4cos(120˚)=2-sqrt3

Consulting the unit circle we find the following:

cos(30˚) = sqrt3/2
sin(30˚) = 1/2
cos(120˚)=-1/2

We can now substitute these values in.

-(sqrt3/2)/{1/2}-4(-1/2)=2-sqrt3

Simplifying we get

-(sqrt3)/{1}+4/2=2-sqrt3=

-sqrt3+2=2-sqrt3

Rearranging we get 2-sqrt3=2-sqrt3