For α, β ∈ (0, π/2), let 3 sin(α + β) = 2 sin(α - β) and a real number k be such that tan α = k tan

Q: Why did we divide by $\cos \alpha \cos \beta$ specifically? Why not just $\cos \alpha$?

That is a fantastic question! We divide by $\cos \alpha \cos \beta$ because our target is to create both $\tan \alpha$ and $\tan \beta$. If we only divided by $\cos \alpha$, we would get $\tan \alpha$ on one side, but we would be left with $\frac{\sin \beta}{\cos \alpha}$, which doesn't help us reach $\tan \beta$. Dividing by the product $\cos \alpha \cos \beta$ is a surgical maneuver that simultaneously transforms $\frac{\sin \alpha}{\cos \alpha}$ into $\tan \alpha$ and $\frac{\sin \beta}{\cos \beta}$ into $\tan \beta$.

Q: Can I use Componendo and Dividendo here? It seems like a ratio problem.

You absolutely can! Componendo and Dividendo is a powerful tool for ratios. If you rewrite the equation as $\frac{\sin(\alpha + \beta)}{\sin(\alpha - \beta)} = \frac{2}{3}$, you can apply the rule to isolate the tangent terms. However, the expansion method we used is often safer because it minimizes the risk of sign errors during the algebraic manipulation. Both paths lead to the same beautiful destination!

Q: Are we allowed to divide by $\cos \alpha \cos \beta$? What if they are zero?

A sharp observation! In mathematics, we must always be wary of division by zero. However, look at the constraints: $\alpha, \beta \in (0, \frac{\pi}{2})$. In the first quadrant, the cosine function is strictly positive and never zero. Therefore, $\cos \alpha$ and $\cos \beta$ are non-zero, making our division perfectly valid and safe.

Question

For $\alpha, \beta \in (0, \frac{\pi}{2})$, let $3 \sin(\alpha + \beta) = 2 \sin(\alpha - \beta)$ and a real number $k$ be such that $	an \alpha = k 	an \beta$. Then the value of $k$ is equal to :

Options:

(A) $-\frac{2}{3}$

(B) $-5$

(C) $\frac{2}{3}$

(D) $5$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given the trigonometric relationship:
$$3 \sin(\alpha + \beta) = 2 \sin(\alpha - \beta)$$

Our objective is to determine the constant $k$ such that the equation takes the form $	an \alpha = k 	an \beta$. To achieve this, we must bridge the gap between the sine functions and the tangent functions.

### The Expansion

We utilize the standard compound angle identities:
$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$
$$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$

Substituting these identities into our original equation, we obtain:
$$3(\sin \alpha \cos \beta + \cos \alpha \sin \beta) = 2(\sin \alpha \cos \beta - \cos \alpha \sin \beta)$$

Distributing the constants across the terms yields:
$$3 \sin \alpha \cos \beta + 3 \cos \alpha \sin \beta = 2 \sin \alpha \cos \beta - 2 \cos \alpha \sin \beta$$

### The Algebraic Dance

Next, we group the like terms by moving all $\sin \alpha \cos \beta$ terms to the left and all $\cos \alpha \sin \beta$ terms to the right:
$$3 \sin \alpha \cos \beta - 2 \sin \alpha \cos \beta = -2 \cos \alpha \sin \beta - 3 \cos \alpha \sin \beta$$

Simplifying both sides of the equation, we arrive at:
$$\sin \alpha \cos \beta = -5 \cos \alpha \sin \beta$$

### Final Transformation

To convert the expression into tangents, we divide both sides of the equation by $\cos \alpha \cos \beta$:
$$\frac{\sin \alpha \cos \beta}{\cos \alpha \cos \beta} = \frac{-5 \cos \alpha \sin \beta}{\cos \alpha \cos \beta}$$

Canceling the common terms, we simplify the expression to:
$$	an \alpha = -5 	an \beta$$

Comparing this result to the target form $	an \alpha = k 	an \beta$, we identify the constant:
**$k = -5$**

Animated Solution for Mathematics - Trigonometry: For $α, β \in (0, \frac{π}{2})$ , let $3 sin (α + β) = 2 sin (α - β)$ and a real number $k$ be such that $tan α = k tan β$ . Then the value of $k$ is equal to :

Select Answer:

Visualized Solution

Given Equation and Target

Compound Angle Formulas

Applying the Formulas

Expanding the Brackets

Grouping $sin α cos β$

Grouping $cos α sin β$

Simplifying the Equation

Creating Tangent Terms

Final Simplification

Comparing with Target

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Expansion

The Algebraic Dance

Final Transformation

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