In a triangle PQR, ∠R = π/2. If tan(P/2) and tan(Q/2) are the roots of ax² + bx + c = 0, a ≠ 0 then

Q: Wait, why do we use $\tan(\frac{P}{2} + \frac{Q}{2})$ instead of just $\tan(P+Q)$?

That is a great observation! The roots of the quadratic equation are given as $\tan(\frac{P}{2})$ and $\tan(\frac{Q}{2})$, not $\tan(P)$ and $\tan(Q)$. To use the information provided by the roots effectively, we must work with the variables we have. By using the half-angle sum, we directly connect the roots to the known value of $\tan(\frac{\pi}{4})$.

Q: How do I know the sum of roots is $-\frac{b}{a}$ and the product is $\frac{c}{a}$?

This is a fundamental result from algebra known as Vieta's formulas. For any quadratic equation $ax^2 + bx + c = 0$, if the roots are $\alpha$ and $\beta$, then $\alpha + \beta = -\frac{b}{a}$ and $\alpha \cdot \beta = \frac{c}{a}$. It is a powerful tool that bridges the gap between the coefficients of a polynomial and its roots.

Q: Is there a specific condition for $a, b, c$ that I should worry about?

The problem specifies $a \neq 0$, which is essential because if $a$ were zero, the equation would not be quadratic. Additionally, since $P$ and $Q$ are angles of a triangle, $\frac{P}{2}$ and $\frac{Q}{2}$ are acute, meaning their tangents are always positive. This ensures the roots are well-defined and the quadratic structure holds.

Question

In a triangle $PQR, \angle R = \pi/2$. If $	an(P/2)$ and $	an(Q/2)$ are the roots of $ax^2 + bx + c = 0, a 
eq 0$ then

Options:

(A) $a = b + c$

(B) $c = a + b$

(C) $b = c$

(D) $b = a + c$

Sigma Percentile · Accepted Answer

### Analyzing the Geometric Foundation

In triangle $PQR$, we are given that the triangle is right-angled at $R$. This serves as our primary anchor.

Since the sum of interior angles in any triangle is $\pi$ radians, we have $\angle P + \angle Q + \angle R = \pi$. Given $\angle R = \frac{\pi}{2}$, it follows that:

$$\angle P + \angle Q = \frac{\pi}{2}$$

This geometric truth is the foundation upon which we will build our algebraic solution.

### The Half-Angle Leap

The provided quadratic equation is $ax^2 + bx + c = 0$, with roots defined as $	an(\frac{P}{2})$ and $	an(\frac{Q}{2})$. To align our geometry with these roots, we divide our previous equation by $2$:

$$\frac{P}{2} + \frac{Q}{2} = \frac{\pi}{4}$$

This step acts as the essential bridge between the triangle's geometry and the algebraic properties of the quadratic equation.

### The Algebraic Engine

Using Vieta's formulas for the quadratic equation $ax^2 + bx + c = 0$, we identify the sum and product of the roots:

$$	an\left(\frac{P}{2}ight) + 	an\left(\frac{Q}{2}ight) = -\frac{b}{a}$$

$$	an\left(\frac{P}{2}ight) \cdot 	an\left(\frac{Q}{2}ight) = \frac{c}{a}$$

These expressions provide the necessary components to utilize trigonometric identities.

### The Grand Synthesis

We apply the tangent addition formula, $	an(A + B) = \frac{	an A + 	an B}{1 - 	an A 	an B}$, where $A = \frac{P}{2}$ and $B = \frac{Q}{2}$:

$$	an\left(\frac{P}{2} + \frac{Q}{2}ight) = \frac{	an(\frac{P}{2}) + 	an(\frac{Q}{2})}{1 - 	an(\frac{P}{2})	an(\frac{Q}{2})}$$

Since $\frac{P}{2} + \frac{Q}{2} = \frac{\pi}{4}$ and $	an(\frac{\pi}{4}) = 1$, we substitute our Vieta's results into the identity:

$$1 = \frac{-\frac{b}{a}}{1 - \frac{c}{a}}$$

### The Final Victory

To solve for the relationship between the coefficients, we simplify the denominator:

$$1 = \frac{-\frac{b}{a}}{\frac{a - c}{a}}$$

Canceling the $a$ from the denominators, we obtain $1 = \frac{-b}{a - c}$. Cross-multiplying yields $a - c = -b$.

Rearranging the terms, we arrive at the final relationship:

**$a + b = c$**

Animated Solution for Mathematics - Trigonometry: In a triangle $P QR, ∠ R = π /2$ . If $tan (P /2)$ and $tan (Q /2)$ are the roots of $a x^{2} + b x + c = 0, a \neq = 0$ then

Select Answer:

Visualized Solution

Visualizing the Right Triangle

Angle Sum Property

Finding $∠ P + ∠ Q$

Transition to Half-Angles

The Quadratic Equation

Sum and Product of Roots

Compound Angle Formula

Substituting the Values

Evaluating the Left Hand Side

Substituting into the Right Hand Side

Simplifying the Denominator

Canceling and Cross-Multiplying

Final Rearrangement

The Sigma Insight: Trigonometric Functions of Compound Angles

Analyzing the Geometric Foundation

The Half-Angle Leap

The Algebraic Engine

The Grand Synthesis

The Final Victory

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Animated Solution for Mathematics - Trigonometry: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle PQR,∠R=π/2. If tan(P/2) and tan(Q/2) are the roots of ax2+bx+c=0,a=0 then