Question

If \( 3 \sin^2 x + 10 \cos x - 6 = 0, 0^\circ<\theta<90^\circ \), then the value of \( \sec x + \csc x + \cot x \) is:

• A. \( 4 - \sqrt{3} \)
• B. \( 2 + \sqrt{3} \)
• C. \( 3 - \sqrt{2} \)
• D. \( 3 + \sqrt{2} \)

Hint:

For such trigonometric equations, manipulate the equation using standard trigonometric identities to express terms in terms of one variable.

Answer 1

The Correct Option is D

We are given the equation \( 3 \sin^2 x + 10 \cos x - 6 = 0 \). First, we solve for \( \sin x \) and \( \cos x \). Rearrange the given equation: \[ 3 \sin^2 x = 6 - 10 \cos x \] \[ \sin^2 x = \frac{6 - 10 \cos x}{3} \] Now, we know that \( \sin^2 x + \cos^2 x = 1 \), so we substitute \( \sin^2 x \) from the above equation: \[ \frac{6 - 10 \cos x}{3} + \cos^2 x = 1 \] Multiply through by 3 to eliminate the denominator: \[ 6 - 10 \cos x + 3 \cos^2 x = 3 \] \[ 3 \cos^2 x - 10 \cos x + 3 = 0 \] This is a quadratic equation in \( \cos x \). Use the quadratic formula to solve for \( \cos x \): \[ \cos x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(3)}}{2(3)} \] \[ \cos x = \frac{10 \pm \sqrt{100 - 36}}{6} \] \[ \cos x = \frac{10 \pm \sqrt{64}}{6} \] \[ \cos x = \frac{10 \pm 8}{6} \] This gives two possible solutions: \[ \cos x = \frac{18}{6} = 3 \quad \text{or} \quad \cos x = \frac{2}{6} = \frac{1}{3} \] Since \( \cos x = 3 \) is not a valid solution (as \( \cos x \) cannot be greater than 1), we have: \[ \cos x = \frac{1}{3} \] Now, use the Pythagorean identity to find \( \sin x \): \[ \sin^2 x = 1 - \cos^2 x = 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \] \[ \sin x = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3} \] Now that we have \( \sin x = \frac{2\sqrt{2}}{3} \) and \( \cos x = \frac{1}{3} \), we can find \( \sec x \), \( \csc x \), and \( \cot x \): \[ \sec x = \frac{1}{\cos x} = \frac{1}{\frac{1}{3}} = 3 \] \[ \csc x = \frac{1}{\sin x} = \frac{1}{\frac{2\sqrt{2}}{3}} = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4} \] \[ \cot x = \frac{\cos x}{\sin x} = \frac{\frac{1}{3}}{\frac{2\sqrt{2}}{3}} = \frac{1}{2\sqrt{2}} \] Now, sum \( \sec x + \csc x + \cot x \): \[ \sec x + \csc x + \cot x = 3 + \frac{3\sqrt{2}}{4} + \frac{1}{2\sqrt{2}} \] Simplify: \[ = 3 + \frac{3\sqrt{2}}{4} + \frac{1}{2\sqrt{2}} = 3 + \sqrt{2} = 3 + \sqrt{2} \] Thus, the correct answer is \( 3 + \sqrt{2} \).