3^0.5tanx-secx-1=0 Find the value of x:

Question

what is the value of x?

Answer 1

tan(x)√3-sec(x)-1=0.

Multiply through by cos(x):

sin(x)√3-1-cos(x)=0,

sin(x)√3=1+cos(x).

cos(x)=2cos²(x/2)-1; sin(x)=2sin(x/2)cos(x/2).

2sin(x/2)cos(x/2)√3=2cos²(x/2),

2cos(x/2)(sin(x/2)√3-cos(x/2))=0,

cos(x/2)=0, x/2=π/2, x=π; or sin(x/2)√3=cos(x/2), tan(x/2)=1/√3, x/2=π/6, x=π/3.

There are many solutions because of the cyclic nature of the trig functions.

tan(x/2)=1/√3, x/2=2πn+(π/6 or 7π/6), which is the same as πn+π/6, x=2πn+π/3;

cos(x/2)=0, x/2=(2n+1)π/2, x=(2n+1)π. n is an integer (positive or negative).

To convert from radians to degrees, replace π by 180°.

In degrees: x=60, 180, 420, 540, 780, etc.