The value of x in the given interval is 5π/6 for the ratio sec x = [tex]-\frac{2\sqrt{3} }{3}[/tex]. This is obtained by using trigonometric identities.
What are the trigonometric identities?
The reciprocal trigonometric identities are:
Sin θ = 1/Csc θ or Csc θ = 1/Sin θ
Cos θ = 1/Sec θ or Sec θ = 1/Cos θ
Tan θ = 1/Cot θ or Cot θ = 1/Tan θ
Calculating the given trigonometric ratio for the given interval:
Given that,
sec x = [tex]-\frac{2\sqrt{3} }{3}[/tex] and the inteval for x is π/2 ≤ x ≤ π
From the reciprocal trigonometric identities, we know that
Sec x = 1/Cos x
So,
1/Cosx = [tex]-\frac{2\sqrt{3} }{3}[/tex]
⇒ Cos x = [tex]-\frac{3 }{2\sqrt{3}}[/tex]
⇒ Cos x = [tex]-\frac{\sqrt{3}\sqrt{3} }{2\sqrt{3} }[/tex]
⇒ Cos x = [tex]-\frac{\sqrt{3} }{2}[/tex]
For finding x value,
x = Cos⁻¹([tex]-\frac{\sqrt{3} }{2}[/tex])
⇒ x = 5π/6
I.e., π/2 ≤ x ≤ π implies π/2 ≤ 5π/6 ≤ π
Therefore, the value of x is 5π/6. So, option b is correct.
Learn more about trigonometric identities here:
https://brainly.com/question/5046810
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