find all solutions in interval [0,2pi]

tan²x = -3/2 secx

This question may look tricky, but its actually quite simple.

First: rearrange the pythagorean identity tan²x+1=sin²x so that it reads tan²x=sin²x-1

Then replace the tan²x in the original problem with the sin²x-1 that you got from rearranging the pythagorean identity

Now the equation should read:
sin²x-1=(-3/2)secx

Next bring the (-3/2)secx to the other side of the equation to make it read:

sin²x+ (3/2)secx -1

Look familiar?

Yes! It is a quadratic. So you should set it equal to zero and factor.

I used the quadratic formula to get the two answers: -2 and .5

Then look on the unit circle and ask yourself: Where is the sec -2? Where is the sec .5?

Answer your questions by finding the points on the unit circle. Use the radians as your answers and don't forget to add 2(pi)n or (pi)n respectively

Let me know if you need more exact instructions

Cori

Im still stuck on the following steps i have no idea what im doing here..........sorry this is the section that ive had sooo much trouble in.

I would do it this way:

tan²x = -3/2 secx

sin²x/cos²x = -3/2cosx

multiply both sides by cosx

sin²x/cosx = -3/2 cross-multiply
2sin²x = -3cosx then by the Pythagorean identity
2(1-cos²x) + 3cosx = 0
2cos²x - 3cosx - 2 = 0
(2cosx+1)(cosx-2) = 0
so cosx = -1/2 or cosx = 2, the last is not possible because cosine is between -1 and 1

then cosx=-1/2 means the angle is in the second or third quadrants.
x = 120º or x = 300º

Done!

To clarify the steps for finding the solutions:

1. Start with the equation tan²x = -3/2 secx.
2. Rewrite tan²x as sin²x/cos²x and secx as 1/cosx.
3. Multiply both sides of the equation by cosx to eliminate the denominators.
4. Simplify the equation to get 2sin²x = -3cosx.
5. Use the Pythagorean identity sin²x = 1 - cos²x to substitute sin²x in the equation.
6. Rearrange the equation to get 2(1 - cos²x) + 3cosx = 0.
7. Simplify further to obtain 2cos²x - 3cosx - 2 = 0.
8. Factorize the quadratic equation to get (2cosx + 1)(cosx - 2) = 0.
9. Set each factor equal to zero and solve for cosx: cosx = -1/2 or cosx = 2.
10. Since cosx cannot be equal to 2 (as cosine values lie between -1 and 1), discard the second case.
11. Solve cosx = -1/2 to find the angles that satisfy this condition.
12. The solutions are x = 120º and x = 300º, which correspond to angles in the second and third quadrants, respectively.

Remember to convert the angles from degrees to radians if the interval is given in radians.

I hope this clears up any confusion you had. Let me know if you have any further questions!