+0

# how to solve

0
283
2

tan x + 4 sec x = 4 , 0<x<360

Guest May 27, 2015

#2
+78741
+5

Ah....Mathcad......you did very well but you gave up a little too quickly, here........!!!!......notice the following factorization....

15sec^2(x)-32sec(x)+17=0

(15sec(x) - 17) (sec(x) -1)  = 0

So either......

sec(x) = 1  which happens at 0 and 360.....but, both of these are out of the requested interval....or....

Here's a graph..........https://www.desmos.com/calculator/fpewbunwdj

Notice that the solution of  28.072°  is "extraneous"......this will frequently happen when we square both sides of an equation......the only  "good" solution in the requested interval occurs at about 331.928°

CPhill  May 28, 2015
Sort:

#1
+122
+5

tan(x)+4sec(x)=4

tan(x)=4(1-sec(x))   Now square both sides to get

tan^2(x)=16(1-sec(x))^2  and using the identity tan^2(x)=sec^2(x)-1 on left hand side

sec^2(x)-1=16(1-sec(x))^2   Expanding brackets and collecting terms gives

15sec^2(x)-32sec(x)+17=0.

Now just solve the quadratic in sec(x) to get the 2 values of the solution. Solution has real roots as b^2-4ac=1024-1020=4,but formula will need to be used as this will not factorise. I am now going for beer.

#2
+78741
+5

Ah....Mathcad......you did very well but you gave up a little too quickly, here........!!!!......notice the following factorization....

15sec^2(x)-32sec(x)+17=0

(15sec(x) - 17) (sec(x) -1)  = 0

So either......

sec(x) = 1  which happens at 0 and 360.....but, both of these are out of the requested interval....or....

Here's a graph..........https://www.desmos.com/calculator/fpewbunwdj

Notice that the solution of  28.072°  is "extraneous"......this will frequently happen when we square both sides of an equation......the only  "good" solution in the requested interval occurs at about 331.928°

CPhill  May 28, 2015

### 8 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details