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If θ and ϕ are the two roots of the quadratic equation 8x² + 22x + 5 = 0, which of the following statements is true?

1. Both sin⁻¹ θ and sin⁻¹ ϕ are real.
2. Both sec⁻¹ θ and sec⁻¹ ϕ are real.
3. Both tan⁻¹ θ and tan⁻¹ ϕ are real.
4. None of these.

Correct answer: Both tan⁻¹ θ and tan⁻¹ ϕ are real.

Solution

The roots θ and ϕ of the quadratic equation are real numbers, and the inverse tangent function (tan⁻¹) is defined for all real numbers, making both tan⁻¹ θ and tan⁻¹ ϕ real.

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