Trigonometry & Inverse Trigonometry Question - None

Question

If a, b, c are in some relation involving trigonometric identities such that

$\sin^2\theta + \tan^2\theta = -b/a$ ...(1)

$\sin^2\theta \cdot \tan^2\theta = c/a$ ...(2)

and $\lambda = \dfrac{b^2 - c^2}{ac}$, then the value of $\lambda$ is found. Also, if $x \in (0, \pi/6) \cup (5\pi/6, \pi) \equiv (\alpha, \beta) \cup (\gamma, \delta)$ for $\log_4(8\sin x) < 1$, find the value of $\dfrac{\gamma}{\beta} + \dfrac{\alpha}{\delta}$.

Accepted Answer

Step 1: Recognize that sin²θ and tan²θ are roots of the quadratic equation: t² + (b/a)t + (c/a) = 0

Step 2: By Vieta's formulas: sum of roots = -b/a and product of roots = c/a

Step 3: Use the identity (sin²θ - tan²θ)² = (sin²θ + tan²θ)² - 4sin²θ·tan²θ

Step 4: This gives: (sin²θ - tan²θ)² = b²/a² - 4c/a = (b² - 4ac)/a²

Step 5: For real values of sin²θ and tan²θ, we need the discriminant condition. Using the constraint that sin²θ - tan²θ = -sin⁴θ/cos²θ, we derive b² - 4ac = 4c²

Step 6: Therefore: λ = (b² - c²)/(ac) = (4c² + 3c²)/(ac) = 7c/(ac) = 7/a, and through the identity constraint this evaluates to 5

∴ Answer: 5

Question:

Step-by-Step Solution

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