SOLUTION: Find all solutions of the equation tan˛x + 2sec˛x = 7, where -<font face="symbol">p</font>/2 < x < <font face="symbol">p</font>/2.

Algebra ->  Trigonometry-basics -> SOLUTION: Find all solutions of the equation tan˛x + 2sec˛x = 7, where -<font face="symbol">p</font>/2 < x < <font face="symbol">p</font>/2.      Log On


   

Question 438002: Find all solutions of the equation tan˛x + 2sec˛x = 7, where -p/2 < x < p/2.
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!


      tan˛x + 2sec˛x = 7

Replace sec˛x by 1 + tan˛x

tan˛x + 2(1 + tan˛x) = 7

Distribute:

  tan˛x + 2 + 2tan˛x = 7

Combine like terms:

          2 + 3tan˛x = 7

Subtract 2 from both sides:

              3tan˛x = 5

Divide both sides by 3

               tan˛x = 5%2F3

Use the principle of square roots

               tan x = %22%22+%2B-+sqrt%285%2F3%29

Rationalize the denominator:

               tan x = %22%22+%2B-+sqrt%28%285%2A3%29%2F%283%2A3%29%29

               tan x = %22%22+%2B-+sqrt%2815%29%2F3  
                        
Since calculators give principle values for the inverse
trig functions, and since -p/2 < x < p/2 is the range for 
tan-1.

One solution is tan-1(sqrt%2815%29%2F3)
                  
which is x = 0.911738291 

The other solution is tan-1(-sqrt%2815%29%2F3)
                  
which is x = -0.911738291
Edwin