SOLUTION: Then angle x satisfies the equation 2tan^2x-5secx-10=0, where x is in the second quadrant. Find the exact value of sec x. I know that the identity of 1+tan^2x=sec^2x may help me

Algebra ->  Trigonometry-basics -> SOLUTION: Then angle x satisfies the equation 2tan^2x-5secx-10=0, where x is in the second quadrant. Find the exact value of sec x. I know that the identity of 1+tan^2x=sec^2x may help me      Log On


   

Question 112355: Then angle x satisfies the equation 2tan^2x-5secx-10=0, where x is in the second quadrant. Find the exact value of sec x.
I know that the identity of 1+tan^2x=sec^2x may help me in this problem, but I am not sure what to do next. Thanks a lot!.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the given equation


Transform the identity to . Now replace with


Distribute


Combine like terms


Now let . So we now get




Let's use the quadratic formula to solve for w:


Starting with the general quadratic

aw%5E2%2Bbw%2Bc=0

the general solution using the quadratic equation is:

w+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve 2%2Aw%5E2-5%2Aw-12=0 ( notice a=2, b=-5, and c=-12)




w+=+%28--5+%2B-+sqrt%28+%28-5%29%5E2-4%2A2%2A-12+%29%29%2F%282%2A2%29 Plug in a=2, b=-5, and c=-12



w+=+%285+%2B-+sqrt%28+%28-5%29%5E2-4%2A2%2A-12+%29%29%2F%282%2A2%29 Negate -5 to get 5



w+=+%285+%2B-+sqrt%28+25-4%2A2%2A-12+%29%29%2F%282%2A2%29 Square -5 to get 25 (note: remember when you square -5, you must square the negative as well. This is because %28-5%29%5E2=-5%2A-5=25.)



w+=+%285+%2B-+sqrt%28+25%2B96+%29%29%2F%282%2A2%29 Multiply -4%2A-12%2A2 to get 96



w+=+%285+%2B-+sqrt%28+121+%29%29%2F%282%2A2%29 Combine like terms in the radicand (everything under the square root)



w+=+%285+%2B-+11%29%2F%282%2A2%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



w+=+%285+%2B-+11%29%2F4 Multiply 2 and 2 to get 4

So now the expression breaks down into two parts

w+=+%285+%2B+11%29%2F4 or w+=+%285+-+11%29%2F4

Lets look at the first part:

x=%285+%2B+11%29%2F4

w=16%2F4 Add the terms in the numerator
w=4 Divide

So one answer is
w=4



Now lets look at the second part:

x=%285+-+11%29%2F4

w=-6%2F4 Subtract the terms in the numerator
w=-3%2F2 Divide

So another answer is
w=-3%2F2

So our possible solutions are:
w=4 or w=-3%2F2



Remember, we let . So or


This means that ===>


and ===>


Now solve for x in each case

Start with the first solution


Take the arccosine of both sides


Take the arccosine of 1%2F4

However, this angle (which is about 75.5 degrees) is in the first quadrant. So it does not satisfy the original conditions



Now move onto the second solution


Take the arccosine of both sides


Take the arccosine of -2%2F3

Converting to degrees, we get about 131.8 degrees, which is in the second quadrant


So the solution is