Can any kind soul help me in the below?
1. If x is obtuse and 2 tan^2 x = 5 sec x +10, find the value of tan x without using calculator.
Answer:
obtuse : 90deg < x< 180 deg
2 tan^2 x = 5 sec x + 10
2 (sec^2 x -1) = 5 sec x + 10
2 (sec^2 x -1) - 5 sec x - 10 = 0
2 sec^2 x -2 - 5 sec x - 10 = 0
(2 sec x + 3)(sec x - 4) = 0
sec x = 3/2 or sec x = 4
sec x^2 = (3/2)^2 or sec x^2 = 4^2
sec x^2 = 9/4 or sec x^2 = 16
(sec x^2) - 1 = 9/4 -1 or (sec x^2) - 1 = 16 - 1
tan x^2 = 5/4 or tan x^2 = 15
tan x = √(5/4) or tan x = √15, -√15
tan x = (√5)/2, -(√5)/2 or tan x = √15, -√15
For x to be obtuse, tan x must be in 2nd quadrant => negative
tan x = -(√5)/2 or tan x = -√15 (rejected)
My question: why is tan x = - √15 rejected?
Hello ppppp77In your working you had (correctly):and you then said (correctly):You then (correctly) rejected because is obtuse and therefore .
But does that mean that is necessarily correct? No, before you can say that, there's one more test to apply, and that is to see whether this value satisfies the original equation; which was:Now the LHS is . But what about on the RHS? Well, using , we get:
and, of course, for , we must take the negative sign, giving
and the RHS then is , not (which is what we wanted).
So doesn't work either.
Tricky little beast, isn't it?
Grandad