If $ 14\sin Y + \cos Y = \sqrt{14} \sin Y $, then find the value of $ an Y $.

Question

If $ 14\sin Y + \cos Y = \sqrt{14} \sin Y $, then find the value of $	an Y $.

Examoogle · Accepted Answer

The correct answer is Option A: $\frac{-14 - \sqrt{14}}{182}$. Rearrange the given linear combination of sine and cosine to isolate tan(Y) by dividing both sides by cos(Y) and grouping the sin(Y) terms.

Step-by-Step Solution & Concepts:
Shortcut Trick: Rearrange the given linear combination of sine and cosine to isolate tan(Y) by dividing both sides by cos(Y) and grouping the sin(Y) terms.

Formula Used: $$\cos(Y) = (\sqrt{14} - 14)\sin(Y)$$, $$	an(Y) = \frac{\sin(Y)}{\cos(Y)}$$

Given: The equation provided is 14 sin(Y) + cos(Y) = sqrt(14) sin(Y).

Steps: Rearrange the Equation: Start with the given equation $14\sin(Y) + \cos(Y) = \sqrt{14}\sin(Y)$ and move the sine terms to one side.. Factor and Isolate: Factor out $\sin(Y)$ on the right side: $\cos(Y) = \sin(Y)(\sqrt{14} - 14)$. Then, isolate $	an(Y)$ using the identity $	an(Y) = \frac{\sin(Y)}{\cos(Y)}$.. Rationalize the Denominator: Multiply numerator and denominator by the conjugate $(\sqrt{14} + 14)$.. Final Verification: The expression $\frac{\sqrt{14} + 14}{-182}$ is equivalent to $\frac{-(14 + \sqrt{14})}{182}$.

Answer: A is correct because the algebraic rearrangement and rationalization of $\frac{1}{\sqrt{14}-14}$ leads to $\frac{-14 - \sqrt{14}}{182}$.

Exam Tip: This problem involves algebraic manipulation of transcendental functions, a skill essential for solving higher-level problems involving trigonometric equations in calculus.

Examoogle · Answer

Option B: $\frac{-14 - \sqrt{14}}{187}$

Examoogle · Answer

Option C: $\frac{-14 - \sqrt{14}}{192}$

Examoogle · Answer

Option D: $\frac{-15 - \sqrt{14}}{182}$

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