Question

How do you use a graphing utility to approximate the solutions of `(1+sinx)/cosx+cosx/(1+sinx)=4` in the interval `[0,2pi)`?

Answer 1

3-sd approximations are 1.05 radian and 5.24 radian, against true transcendental values 1.047197.. and 5.2359877...

Explanation:

The first graph is for `x in [0, 2pi]`, for locating the solutions as x-

intercepts. There are two and they are close to 1 radian and 5.2 radian.

graph{(1+sinx)/cosx+cosx/(1+sinx)-4 [0, 6.28, -5, 5]}

The narrowing of ranges for x and y, around `x = 1_+` reveals x =

1.05 radian as 3-sd approximation to the smaller zero.ti

graph{(1+sinx)/cosx+cosx/(1+sinx)-4 [1, 1.1, -5, 5]}

Narrwing ranges around x = 5.2 reveals the other zero as 5.24 for 3-sd.

graph{(1+sinx)/cosx+cosx/(1+sinx)-4 [5.2, 5.3, -5, 5]}

Algebraically, cos x= 1/2 and the solutions are `pi/3=1.0472`, for 5-

sd approximation and `5/3pi=5.2360`, nearly.

Graphs have limits for scaling precision.