College Trigonometry: Suppose that 90 < x < 180 and tan2(x) = -1/3

[misty_falls]

Hi. I have some trigonometry problems I am stuck with.

1.) Suppose that 90 < x < 180 and tan2(x) = -1/3. What is the exact value of tan(x)?
So for this I am stuck with a quadratic equation of tan^2(x) - 6tan(x) - 1 = 0. I don't know what step next to do to find the value of tan x.

2.) Find the raidus of a circle if we know that a sector of 38° and has area of 7ft squared.

 

[Ishuda]

First, the quadratic equation. Let u = tan(x) and you have
u² - 6 u - 1 = 0
To choose the proper sign to use for the solution, consider the sign of tan(x).

If you had a sector of 180° you would have half a circle ( = 180/360 ) so the area of the sector would be (1/2) the area of the circle or \(\displaystyle \frac{1}{2} \pi\space r^2\). Suppose that equaled \(\displaystyle \frac{1}{2} \pi\). What would r be?

 

[HallsofIvy]

What, exactly, do you mean by "tan2(x)"? Is it \(\displaystyle tan^2(x)\) or \(\displaystyle tan(2x)\)?

 

[misty_falls]

What, exactly, do you mean by "tan2(x)"? Is it \(\displaystyle tan^2(x)\) or \(\displaystyle tan(2x)\)?

Yes Ivy. It's tan(2x) = -1/3 and I was asked to find the value of tan(x). I'm stuck with the quadratic and don't know what to do next.
I remember something that I have to square root to get rid of the square but I have no idea how.

 

[Ishuda]

Yes Ivy. It's tan(2x) = -1/3 and I was asked to find the value of tan(x). I'm stuck with the quadratic and don't know what to do next.
I remember something that I have to square root to get rid of the square but I have no idea how.

The two solutions to the zeros of a quadratic equation
a u² + b u + c = 0
are
\(\displaystyle u_1 = \frac{-b - \sqrt{b^2 - 4 a c}}{2 a}\)
and
\(\displaystyle u_2 = \frac{-b + \sqrt{b^2 - 4 a c}}{2 a}\)

Thus tan(x) is either u₁ or u₂. Note that one of the solutions for your problem will be positive and the other will be negative. What is the sign of the tangent when 90 < x < 180?

Since they want the exact answer, it will not be possible to get rid of the square root.

 

[misty_falls]

Thank you Ishuda but what does a value as?

 

[HallsofIvy]

You have been told exactly what to do. Can you not do the arithmetic?

 

[Ishuda]

Thank you Ishuda but what does a value as?

a = 1
b = -6
c = -1

 

[misty_falls]

You have been told exactly what to do. Can you not do the arithmetic?

You don't have to be so condescending.